Formation of Giant Planet Satellites

Konstantin Batygin1 and Alessandro Morbidelli21 Division of Geological and Planetary Sciences California Institute of Technology, Pasadena, CA 91125, USA

2 Laboratoire Lagrange, Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, CS 34229, F-06304 Nice, FranceReceived 2020 January 29; revised 2020 April 9; accepted 2020 April 13; published 2020 May 18

Abstract

Recent analyses have shown that the concluding stages of giant planet formation are accompanied by thedevelopment of a large-scale meridional flow of gas inside the planetary Hill sphere. This circulation feeds acircumplanetary disk that viscously expels gaseous material back into the parent nebula, maintaining the system ina quasi-steady state. Here, we investigate the formation of natural satellites of Jupiter and Saturn within theframework of this newly outlined picture. We begin by considering the long-term evolution of solid material, anddemonstrate that the circumplanetary disk can act as a global dust trap, where s•∼0.1–10 mm grains achieve ahydrodynamical equilibrium, facilitated by a balance between radial updraft and aerodynamic drag. This processleads to a gradual increase in the system’s metallicity, and eventually culminates in the gravitational fragmentationof the outer regions of the solid subdisk into ~ 100 km satellitesimals. Subsequently, satellite conglomerationensues via pair-wise collisions but is terminated when disk-driven orbital migration removes the growing objectsfrom the satellitesimal feeding zone. The resulting satellite formation cycle can repeat multiple times, until it isbrought to an end by photoevaporation of the parent nebula. Numerical simulations of the envisioned formationscenario yield satisfactory agreement between our model and the known properties of the Jovian and Saturnianmoons.

Unified Astronomy Thesaurus concepts: Satellite formation (1425); Galilean satellites (627); Saturniansatellites (1427)

1. Introduction

With the tally of confirmed extrasolar planets now firmly inthe thousands (Thompson et al. 2018), the feeling ofastonishment instigated by the disparity between orbitalarchitectures that comprise the galactic planetary census andthat of our own solar system is difficult to resist. Indeed, thewidespread detection of planets that complete their orbitalrevolutions in a matter of days and have masses on the order of10–100 ppm of their host stars3 has inspired a large-scalereimagination of the dominant physical processes that makeup the standard model of planet formation (Morbidelli &Raymond 2016; Johansen & Lambrechts 2017). In hindsight,however, the prevalence of such planetary architectures wasalready foreshadowed by Galilleo’s discovery of Jupiter’sregular satellites and Huygens’ subsequent discovery of Titanin orbit around Saturn, some four centuries ago.

Characterized by orbital periods that range from approxi-mately two days (Io) to slightly in excess of two weeks(Callisto and Titan), as well as cumulative masses that add upto about 0.02% of their host planets, both the physical andorbital machinery of giant planet satellites eminently reflect theproperties of typical planetary systems found throughout theGalaxy (Laughlin & Lissauer 2015). Even the intrasystemuniformity inherent to the demographics of sub-Jovianextrasolar planets (Millholland et al. 2017; Weiss et al. 2018)is aptly reproduced in the Galilean ensemble of moons. Theensuing possibility that this similarity may point to a deeperanalogy between conglomeration pathways of solar systemsatellites and short-period exoplanets has not eluded theliterature (Kane et al. 2013; Ronnet & Johansen 2020).Nevertheless, it is intriguing to notice that while the pursuit

to quantify the formation of extrasolar super-Earths hasreceived considerable attention over the course of the recentdecades (see, e.g., the recent works of Bitsch 2019; Izidoroet al. 2019; Liu et al. 2019; Rosenthal & Murray-Clay 2019;Kuwahara & Kurokawa 2020; Poon et al. 2020 and thereferences therein), a complete understanding of the formationof the solar system’s giant planet satellites themselves remainsincomplete (Canup & Ward 2009; Miguel & Ida 2016; Ronnet& Johansen 2020). Outlining a new theory for the theirconglomeration is the primary purpose of this paper.Much like the prevailing narrative of the solar system’s

formation, the theory of satellite formation traces its roots to thenebular hypothesis (Kant 1755; Laplace 1796). Both the Galileanmoons, as well as Titan, are generically thought to have originatedin dissipative disks of gas and dust that encircled Jupiter andSaturn during the first few millions of years of the solar system’slifetime. Within these circumplanetary disks, dust is envisioned tohave solidified into satellitesimals through some physicalmechanism, and assisted by gravitational and hydrodynamicprocesses, the satellitesimals eventually grew into the moons weobserve today (Peale 1999). The devil, however, is in the details,and broadly speaking, current theories of natural satelliteformation fall into two categories: the minimum mass modeland the gas-starved model. Let us briefly review the qualitativecharacteristics of these theories.

1.1. Existing Models

The minimum mass model, first proposed by Lunine &Stevenson (1982), posits that the cumulative present-day massof the satellites approximately reflects the primordial budget ofsolid material that was entrained within the giant planets’circumplanetary disks. Correspondingly, under the assumptionof nearly solar metallicity, the minimum mass model entails

The Astrophysical Journal, 894:143 (23pp), 2020 May 10 https://doi.org/10.3847/1538-4357/ab8937© 2020. The American Astronomical Society. All rights reserved.

3 For a sunlike star, this mass ratio corresponds to M∼3–30 M⊕ planets.

1

disks that were only a factor of ∼50 less massive than the giantplanets themselves, and therefore verge on the gravitationalstability limit of nearly Keplerian systems (Safronov 1960;Toomre 1964). Generically, this picture is characterized byhigh disk temperatures and very short satellite conglomerationtimescales.

A somewhat more modern extension of this model wasconsidered by Mosqueira & Estrada (2003a, 2003b), whoproposed a circumplanetary nebula that is broken up into a denseinner component and a more tenuous outer region. Within thismodel, Galilean satellites are imagined to form from ∼1000 kmseeds by accretion of smaller inward-drifting satellitesimals, butonly the inner three embryos—which form in the dense inner disk—suffer convergent orbital evolution that locks them into theLaplace resonance. Meanwhile, the nebula envisioned byMosqueira & Estrada (2003a, 2003b) is almost perfectly quiescentby construction, such that the steep break in the surface densitycan persist for the entire lifetime of the system, acting as aneffective barrier that halts Callisto’s disk-driven orbital decay.4

The ideas outlined by Mosqueira & Estrada (2003a, 2003b)were explored in a systematic manner by Miguel & Ida (2016).Varying an impressive range of parameters within theirsimulations, Miguel & Ida (2016) have convincingly demon-strated that even if one allows for drastic tailoring of thephysical state of the system (e.g., ad-hoc modification of themigration timescale as well as the solid-to-gas ratio by ordersof magnitude), the formation of a satellite system thatresembles the real Galilean moons remains exceptionallyunlikely within the context of the minimum mass model.

The gas-starved model, put forward by Canup & Ward(2002) proposes a markedly different scenario. In this picture,the circumplanetary disk is not treated as a closed system, andis assumed to actively interact with its environment, con-tinuously sourcing both gaseous and solid material from thesolar nebula. Correspondingly, as solid material is brought intothe system, it is envisioned to accrete into large bodies, which—upon becoming massive enough—experience long-rangeinward orbital decay due to satellite–disk interactions. In thismanner, the circum-Jovian disk considered by Canup & Ward(2002) is not required to retain the full mass budget of theGalilean satellites at any one time and can instead remain in aquasi-steady state with relatively low density.

A key advantage of the gas-starved model is the combinationof a comparatively long satellite migration timescale and asufficiently low disk temperature for effective growth of icybodies. Because objects that reach the inner edge of the disk areassumed to get engulfed by the planet,5 the concurrentoperation of these two processes (i.e., conglomeration andmigration) determines a characteristic steady-state mass scaleof the satellites that occupy the disk. Impressively, within theframework of the Canup & Ward (2002, 2006) scenario, thisquantity evaluates to approximately one-ten-thousandth of thehost planet mass, in agreement with observations.

Despite the numerous successes of the gas-starved model inexplaining the basic architecture of the solar system’spopulation of natural satellites, recent progress in theoreticalmodeling of circumplanetary disk hydrodynamics (Tanigawaet al. 2012; Morbidelli et al. 2014; Szulágyi et al. 2014, 2016;Lambrechts et al. 2019) has revealed a number of intriguingnew challenges pertinent to the formation of giant planetsatellites. The most pivotal of the fledging issues concerns theaccumulation of a sufficient amount of solid material within thedisk. In particular, both numerical simulations (e.g., Tanigawaet al. 2012; Morbidelli et al. 2014) as well as directobservations (Teague et al. 2019) show that gas is deliveredinto the planetary Hill sphere via meridional circulation that issourced from a region approximately one pressure scale heightabove the midplane of the parent circumstellar nebula. Becauseof preferential settling of solids to the midplane (e.g.,Lambrechts & Johansen 2012), solid material that enters theplanetary region is both scarce (implying a sub-solar metallicityof the gas) and small enough (i.e., much smaller than ∼0.1 mm)to remain suspended at large heights within the parent nebula.This picture stands in stark contrast with the scenario of Canup& Ward (2002, 2006), where 1 m satellitesimals areenvisioned to get captured by the circumplanetary disk.A second problem inspired by the emerging view of giant

planet-contiguous hydrodynamics concerns the formation andgrowth of satellitesimals themselves. That is to say, neither theprocess by which incoming dust gets converted into satellitebuilding blocks nor the mechanism through which these soliddebris coalesce within strongly sub-Keplerian circumplanetarydisks to form the satellites is well understood. Finally, argumentsbased upon the accretion energetics of the giant planet envelopesas well as considerations of angular momentum transport duringthe final stages of planetary growth suggest that the primordialmagnetospheres of Jupiter and Saturn could have effectivelytruncated their circumplanetary disks (see for example, Batygin2018). As pointed out by Sasaki et al. (2010), the resulting cavitieswithin the circumplanetary nebulae could feasibly disrupt theaccretion-migration-engulfment cycle envisioned by Canup &Ward (2006).

1.2. This Work

Motivated by the aforementioned developments, in thiswork, we re-examine the dynamical states of circumplanetarydisks during the giant planets’ infancy and propose a newmodel for the conglomeration of giant planet satellites. We startfrom first principles, and throughout the manuscript, consis-tently focus on the characterization of the dominant physicalprocesses, attempting not to prioritize any specific scenario forthe evolution of the satellites. Nevertheless, as we demonstratebelow, consideration of the basic gravito-hydrodynamicmachinery of the protosatellite disks naturally lends itself to aself-consistent picture for the origins of Jovian and Kronianmoons.6

Put succinctly, our model envisions the gradual accumula-tion of icy dust in a vertically fed decretion disk7 that encirclesa newly formed giant planet. Buildup of solid material within

4 Within the broader framework of satellite–disk interactions, a large surfacedensity gradient leads to a dramatic enhancement in the corotation torque. Inturn, this effect preferentially pulls the migrating object into the region ofhigher density (Masset et al. 2006; Paardekooper & Johansen 2018). Therefore,the surface density jump envisioned by Mosqueira & Estrada (2003a, 2003b) islikely to operate as a reversed planet trap, briefly accelerating—instead ofhalting—Callisto’s orbital decay.5 We note that contrary to this assumption, Sasaki et al. (2010) argue that theplanetary magnetosphere may effectively truncate the circumplanetary disk,halting the inward migration of satellites at the inner edge.

6 The satellite systems of Uranus and Neptune are beyond the scope of ourstudy, as they likely have a distinct origin from the scenario considered herein(see, e.g., the recent work of Ida et al. 2020).7 Contrary to accretion disks—where long-term viscous evolution leads to thegradual sinking of nebular material toward the central object—decretion disksare systems where gas and dust are slowly expelled outwards.

2

The Astrophysical Journal, 894:143 (23pp), 2020 May 10 Batygin & Morbidelli

the circumplanetary nebula is driven by a hydrodynamicalequilibrium, which arises from a balance between viscousoutflow of the gas along the disk’s midplane (that drives dustoutward) and sub-Keplerian headwind (that saps the dust of itsorbital energy). The cancellation of these two effects allowsparticles with an appropriate size-range to remain steady withinthe system. As the cumulative mass of solids within the diskslowly grows, the dust progressively settles toward themidplane of the circumplanetary disk under its own gravity.Eventually, gravitational collapse ensues, generating largesatellitesimals that are comparable in size to Saturn’s smallmoons.

Mutual collisions among satellitesimals facilitate oligarchicgrowth, generating satellite embryos. Upon reaching a criticalmass—determined by an approximate correspondence betweenthe timescale for further accretion and the orbital migrationtime—newly formed satellites suffer long-range orbital decay,which terminates when the bodies reach the vicinity of thedisk’s magnetospheric cavity. Owing to continuous aerody-namic damping of the satellitesimal velocity dispersion, thesatellite conglomeration process can repeat multiple times butnecessarily comes to a halt after the photoevaporation frontwithin the circumstellar nebula reaches the giant planet orbit. Aqualitative sketch of our model is presented in Figure 1.

In the remainder of the paper, we spell out the specifics ofour theory. In Section 2, we outline the model of thecircumplanetary disk. The dynamics of dust within the modelnebula are discussed in Section 3. Section 4 presents acalculation of satellitesimal formation. Oligarchic growth of thesatellite embryos and orbital migration are considered inSection 5. Section 6 presents a series of numerical experimentsthat quantify the conglomeration of the satellites themselves, aswell as the formation of the Laplace resonance. We concludeand discuss the implications of our proposed picture inSection 7.

2. Model Circumplanetary Disk

The stage for satellite formation is set within the circumpla-netary disk, and the construction of a rudimentary model for itsstructure is the foundational step of our theory. Recentadvances in high-resolution numerical hydrodynamics simula-tions of quasi-Keplerian flow around giant planets (Tanigawaet al. 2012; Szulágyi et al. 2016) have revealed a nuancedpattern of fluid motion that develops in the vicinity of amassive secondary body when it is embedded within acircumstellar nebula.

First, as a consequence of gravitational torques exerted bythe planet on its local environment, the gas surface densitydrastically diminishes in the planet’s orbital neighborhood,clearing a gap within the circumstellar disk (Crida et al. 2006;Fung & Chiang 2016). Within the gap itself, a meridionalcirculation ensues close to the planet, such that gaseousmaterial rains down toward the planet in a quasi-vertical matter,from an altitude of approximately one disk pressure scaleheight8 (Tanigawa et al. 2012; Morbidelli et al. 2014; Szulágyiet al. 2016). Owing to angular momentum conservation, this

material spins up as it freefalls, consolidating into a sub-Keplerian circumplanetary disk. This disk spreads viscously,decreating outwards, such that the constituent gas gets recycledback into the circumstellar nebula (Figure 1).Importantly, the results of the aforementioned hydrodyna-

mical calculations have shown a remarkable degree ofagreement with contemporary observations. In particular,resolved disk gaps—routinely attributed to dynamical clearingby giant planets—have become a staple of both submillimetercontinuum maps, as well as scattered light images ofprotoplanetary nebulae (Isella & Turner 2018; Zhang et al.2018 and the references therein). Moreover, the recentdetection of the first circumplanetary disk in the PDS 70system (Isella et al. 2019) as well as the characterization ofmeridional circulation of gas through observations of 12COemission in the HD 163296 system (Teague et al. 2019) lendfurther credence to the qualitative picture outlined above.Swayed by the emergent census of circumplanetary disk

simulations and observations, here, we adopt an analytic modelfor a constant M decretion disk as our starting point. Thismodel was first developed within the context of Be stars by Leeet al. (1991) and is related to the routinely utilized constant Maccretion disk model (Armitage 2010). However, the differ-ences between the decretion and accretion models aresufficiently subtle that it is worthwhile to sketch out themodel’s derivation.We begin by recalling the continuity equation for the surface

density, Σ, of the circumplanetary disk (Pringle 1991),

¶ S¶

+¶¶

S = rt r

r v , 1r( ) ( )

where r is the planetocentric distance, and vr is the radialvelocity of the fluid. The rhs of the above expression is a d-function source term that is only nonzero at the inner and outerboundaries of the disk, which we take to be the radius of themagnetospheric cavity, RT, and the Hill radius, RH, respec-tively. In other words, despite the fact that the meridional flowspans a broad range in r, here, we adopt the simplifyingassumption that the vertical flux of material into the disk islocalized to <r RT. Explicitly, the two aforementioned values—which serve as the confines of our model—are calculated asfollows:

zm

m= =

R

M MR a

M

M2 3, 2T

7

0

4

2

1 7

H

1 3

( )◦

◦⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

where ζ is a dimensionless constant of order unity, μ0 is thepermeability of free space, while a, M◦, and μ are the planet’sheliocentric semimajor axis, mass, and magnetic moment,respectively (Mohanty & Shu 2008). For definitiveness, weadopt Jovian parameters, noting that the Hill radii of Jupiterand Saturn are approximately one-third and one-half of anastronomical unit, respectively, while for system parametersrelevant to the final stages of runaway accretion, the disk’smagnetospheric truncation radius evaluates to ~R R5T Jup—avalue marginally smaller than Io’s present-day semimajor axis(Batygin 2018).In steady state (¶ ¶ t 0), the first term of Equation (1)

vanishes, such that in the region of interest ( < <R r RT H), the

8 It is worth noting that recent high-resolution numerical experiments(Lambrechts et al. 2019) suggest that for a Jupiter-mass object, such circulationonly operates for a sufficiently low mass-accretion rate, which translates to anepoch when the bulk of the planetary mass has already been acquired. Withinthe context of our model, this means that our envisioned scenario is set towardthe last approximately million years of the solar nebula’s lifetime.

3

The Astrophysical Journal, 894:143 (23pp), 2020 May 10 Batygin & Morbidelli

solution is simply given by

pS = =r v

M

2const. 3r ( )

This expression establishes a connection between the surfacedensity and radial fluid velocity at all relevant radii,parameterized by the rate at which mass flows through thesystem, which we set to = -M M0.1 MyrJup

1 . We emphasizethat this value of M is low compared with the characteristicmass-accretion rate of nebular material onto T-Tauri stars(which is closer to ~ -

M M10 MyrJup1 ) and is appropriate

only for the concluding stage of the circumstellar disk’sevolution, when our model is envisioned to operate.

The continuity equation for angular momentum within thedisk is written as follows (Lynden-Bell & Pringle 1974):

n¶ S W

¶+

¶ S W¶

=¶¶

S¶ W¶

rr

t

r v

r rr

r, 4r

2 33( ) ( ) ( )⎜ ⎟⎛

⎝⎞⎠

where ν is the viscosity, and W = M r3◦ is the Keplerian

orbital frequency (i.e., mean motion) of the disk material. As

before, the steady-state assumption eliminates the leading termof Equation (4), and upon substituting the definition of M andtaking the derivative, we rewrite the momentum continuityequation as follows (Lee et al. 1991):

p¶

¶+ =

r

r

M M

r40, 5

2( ) ( )◦

where n= - S ¶W ¶ r r is identified as the verticallyintegrated viscous stress tensor.Equation (5) is readily solved for as a function of r, upon

specification of a single boundary condition. To this end, weassume that the viscous torque vanishes at the outer edge of thedisk, such that = 0 at =r RH. We then have

p= W -

M R

r21 . 6H ( )

⎛⎝⎜

⎞⎠⎟

Noting that ¶W ¶ = - Wr r3 2( ) and canceling the depend-ence on W in the above expression, we obtain the surface

Figure 1. A qualitative sketch of our model. A giant planet of mass ~ -M M10 3◦ is assumed to have cleared a gap within its parent nebula, resulting in a steady-state

azimuthal/meridional circulation of gas within the planet’s Hill sphere. As nebular material overflows the top of the gap (at an altitude of approximately one pressurescale height above the midplane), it freefalls toward the planet—a process we parameterize by an effective mass-flux ~ -M M0.1 Myr 1

◦ . Due to the conservation ofangular momentum, this material spins up and forms a circumplanetary disk. The inner edge of this disk is truncated by the planetary magnetic field, B∼1000 G, at aradius ~R R5T Jup. Owing to (magneto-)hydrodynamic turbulence within the circumplanetary disk—parameterized via the standard α∼10−4 prescription—the diskspreads viscously and settles into a steady pattern of decretion back into the circumstellar nebula. A balance of viscous heating and radiative losses within the diskdetermines the system’s aspect ratio, h/r∼0.1. The associated pressure support gives rise to sub-Keplerian rotation of the gas, such that the radial and azimuthalcomponents of the circumplanetary flow are ~ >-v v10 0r

5K and ~ <fv v v0.99 K K, respectively. For a critical dust size s•∼0.1–10 mm, aerodynamic energy loss

from the sub-Keplerian headwind exactly cancels the energy gain from the radial wind, trapping the incoming dust within the disk. Through this process, dustaccumulates within the system, and disk metallicity, , grows in time. This gradual enhancement of the dust-to-gas ratio causes the solid subdisk to settle ever closerto the midplane, and once its scale height, h•, falls below the threshold of gravitational stability (Q•1), the outer regions of the solid subdisk fragment into a swarmof m∼1019 kg satellitesimals. Accretion of satellite embryos proceeds through pair-wise collisions, aided by gravitational focusing. Conglomeration is terminatedonce a satellite grows sufficiently massive to raise appreciable wakes within the circumplanetary disk. At this point, long-range disk-driven migration ensues, usheringthe newly formed object toward the magnetospheric cavity.

4

The Astrophysical Journal, 894:143 (23pp), 2020 May 10 Batygin & Morbidelli

density profile of the disk (Figure 2),

p nS = -

M R

r31 . 7H ( )

⎛⎝⎜

⎞⎠⎟

We note that herein lies an important difference betweenconstant M accretion and decretion disks. For an accretion diskwhere M is negative, the relation

pn

= -¶ S

¶M

r r

r

r2

38

( ) ( )

is satisfied by νΣ=const. This, however, cannot hold true inprinciple for a decretion disk. That is, since radial motion of thedisk material is facilitated by viscous spreading, Equation (8)necessitates that νΣ must be a function that decays moresteeply in radius than r , for M (and by extension, vr) to bepositive.

In order to complete the specification of the problem, wemust define the functional form of the viscosity, and to do so,we adopt the standard Shakura & Sunyaev (1973) αprescription, setting

n a a= = Wc h h , 9s2 ( )

where m= = Wc k T hs b is the isothermal speed of sound,and h is the pressure scale height. Of course, the value of αitself is highly uncertain. Nevertheless, we note that recentresults from the DSHARP collaboration (Dullemond et al.2018) report lower limits on turbulent viscosity withincircumstellar disks that translate to α∼10−4. Following thiswork, here we set α=10−4 but note that the surface densityprofile itself only depends on the ratio of aM , implying aconsiderable degeneracy between two poorly determinedquantities.

Assuming that the circumplanetary disk is “active”, the disktemperature is determined by an energy balance between viscousheat generation within the nebula, p n= S W+Q r9 8 2 2( ) ,

and blackbody radiative losses from its surface, which, for anoptically thin disk, have the simple form p s=-Q r T4 4

(Armitage 2010). In turn, recalling the proportionality betweentemperature and the speed of sound, this equilibrium determinesthe geometrical aspect ratio =h r c vs K of the disk,

m p s= -

h

r

k

M

M M r R

r

3

81 . 10b H

1 4

( )◦

◦⎛⎝⎜

⎛⎝⎜

⎞⎠⎟

⎞⎠⎟

We admit that our assumption of an optically thin nebula is asimplifying one and caution that it can only be justified if thesystem’s budget of micron-sized dust is low. However, becausemicron-sized dust is, in general, very tightly coupled to the gas,it is unlikely that it can ever accumulate in a steady-statedecretion disk, implying that our optically thin and isothermalassumption may be defensible.Quantitatively, the above expression evaluates to h/r∼0.1

throughout the circumplanetary nebula, implying a relativelythick, almost unflared disk structure (Figure 3). We further notethat the aspect ratio only exhibits a very weak dependence on M ,and no explicit dependence on α, insinuating a pronounced lackof sensitivity to poorly constrained parameters. With theexpression for the disk scale height specified, the midplane gasdensity can be calculated in the usual manner: r p= S h2( ).Of course, the underlying assumptions of Equation (10) are

only sensible if viscous heating dominates over planetaryirradiation in the region of interest. Quantitatively, this physicalregime is appropriate if the ratio between radiative and viscous

Figure 2. Surface density profile of our model circumplanetary disk. The exactsolution (Equation (7)), corresponding to a constant-M decretion a-model, isshown with a solid purple line. An index −5/4 power-law fit to this solution(Equation (12)) is depicted by the dashed dark orange line. The referencesurface density Σ0=4000 g cm−2 and reference radius =r R0.10 H are markedwith thin dotted lines. The (assumed vertically isothermal) temperature profileof the disk is represented at the top of the figure with a color bar. Importantly,within the context of our model, temperatures are sufficiently low for icecondensation to ensue beyond rr0. The inner edge of the disk is determinedby the size of the magetospheric cavity (Equation (2)), which we take tobe ~R R5T Jup.

Figure 3. Aspect ratio of the circumplanetary nebula. Within the framework ofour model, the vertical thickness of the gaseous disk merely reflects itstemperature structure (via = µh r c v Ts K ). Because viscous energydissipation dominates over planetary irradiation (Equation (11)), the temper-ature profile itself is determined by equating turbulent heating within the disk toblackbody radiative losses at its surface. The top panel shows the exact solutionfor the gaseous component of the nebula (Equation (10)) with a solid purpleline. For our purposes, it suffices to ignore minor variations in h/r with orbitalradius and envision the disk as having a constant aspect ratio. Our adoptedvalue of h/r=0.1 is shown with the dashed dark orange line in the top panel.The aspect ratio of the dust layer—computed in the massless tracer-particlelimit with Sc of unity (Equation (22))—is shown on the top panel with a blackcurve. The bottom panel shows the aspect ratio of the solid subdisk, h•/r, for avariety of disk metallicities, accounting for energetic suppression of the dustdisk’s thickness (Equation (29)). Note that once the energetic limitation of theturbulent stirring of the dust layer is taken into consideration, h•/r=h/r.Moreover, it is worthwhile to note that µ h r 1• and that for 0.1,h•/r10−3.

5

The Astrophysical Journal, 894:143 (23pp), 2020 May 10 Batygin & Morbidelli

heating

s=

-

T R

M M R r

16

9 111

4 3

H( )( )◦ ◦

◦

is significantly smaller than unity. Indeed, for planetaryparameters of T◦=1000 K and =R R2 Jup◦ , < 1 forr R 2H , albeit only by a factor of a few. This means that

even though irradiation from the central planet does notdominate the disk’s thermal energy balance, it can contribute anotable correction (e.g., minor flaring) to the aspect ratio profile(10), especially in the outer regions of the circumplanetarynebula. Here, however, we neglect this technicality to keep themodel as simple as possible.

While Equations (7) and (10) provide exact solutions for thesurface density and aspect ratio profiles of the model nebula,much of the literature on astrophysical disks is built around theconsideration of power-law models, and it is illustrative tomake the connection between this simplified description andthe more self-consistent picture outlined above. Thus, adopting

=r R0.10 H as a reference radius, we find that our model can becrudely represented by the fit

S » S »r

r

h

r0.1, 120

05 4

( )⎜ ⎟⎜ ⎟⎛⎝

⎞⎠

⎛⎝

⎞⎠

where the reference surface density at r=r0 is Σ0≈4000 g cm−2. For comparison, we note that the aboveexpression corresponds to a protosatellite nebula that is slightlysteeper than a Mestel (1963)–type Σ∝r−1 disk, while beingmarginally shallower than a Hayashi-type minimum mass solarnebula (Weidenschilling 1977; Hayashi 1981). These rudimen-tary profiles are shown alongside Equations (7) and (10) inFigures 2 and 3.

A subtle but important consequence of radial pressuresupport, ∂ P/∂ r, within the circumplanetary disk is the sub-Keplerian rotation of the gas. A conventional way toparameterize the degree to which the gas azimuthal velocity,vf, lags the Keplerian velocity, vK, is to introduce the factor(e.g., Armitage 2010)

hr

= -¶¶

= ~ -

r

M

P

r

h

r

1

2

13

810 , 13

2 22( ) ( )

◦⎜ ⎟⎛⎝

⎞⎠

where the numerical factor on the rhs corresponds to the power-law surface density profile in Equation (12). Accordingly, boththe azimuthal and radial velocity of the fluid within thecircumplanetary nebula are now defined and have the form

h

p p

= - » - <

=S

»S

>

fv v vh

rv

vM

r

M

r

r

r

1 2 113

8

2 20.

14

r

K K

2

K

0 0

5 4( )

⎜ ⎟⎛⎝⎜

⎛⎝

⎞⎠

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

With the specification of the model circumplanetary diskcomplete, let us now examine the evolution of solid dustembedded within this nebula.

3. Dust Dynamics

A rudimentary precondition that must be satisfied within thecontext of giant planet satellite formation theory is theaccumulation of a sufficient amount of solid material withinthe circumplanetary disk. As already mentioned in theintroduction, however, this basic issue of accumulating thenecessary high-metallicity mass budget to form the satellites( ´ -M M 2 10•

4◦ , where M• refers to the total mass of

solids within the disk) poses a formidable problem (Ronnet &Johansen 2020). In fact, the difficulty in capturing icy androcky matter from the circumstellar nebula is two-fold.On one hand, the majority of small grains that drift toward

the planet’s semimajor axis by way of aerodynamic drag getsshielded away from the planet’s orbital neighborhood due to alocal pressure maximum that develops in the planet’s vicinityfor ÅM M30◦ (Lambrechts et al. 2014). More specifically,the high-resolution numerical simulations of Weber et al.(2018) and Haugbølle et al. (2019) suggest that only grainssmaller than s0.1 mm are sufficiently well coupled to thegas to remain at a high enough altitude in the circumstellar diskto bypass the midplane pressure barrier and enter into theplanetary Hill sphere together with the meridional circulation.Even so, in light of their near-perfect coupling to the gas, it isa priori unclear how such small grains can become sequesteredin the circumplanetary disk, instead of getting expelled backinto the circumstellar nebula, together with the decretion-ary flow.On the other hand, direct injection of planetesimals into the

circumplanetary disk appears problematic from an energeticpoint of view. That is, calculations of Estrada et al. (2009) andMosqueira et al. (2010) suggest that planetesimals that aresuccessfully captured around the planet inevitably experiencelarge-scale ablation (see also the recent work of Ronnet &Johansen 2020). Accordingly, small grains generated from theprocess of planetesimal evaporation likely suffer the same fateas grains injected by the meridional circulation. Meanwhile,larger fragments—even if extant—are likely to rapidly spiralonto the planet, due to interaction with a strong headwindgenerated by the appreciably sub-Keplerian flow of thecircumplanetary gas (Weidenschilling 1977).In this section, we outline how this problem is naturally

circumvented within the framework of our model. We begin bywriting down the well-studied equations of motion for a solidparticle of radius s• in orbit of the central planet, whichexperiences aerodynamic drag in the Epstein regime, arisingfrom the marginally sub-Keplerian flow of circumplanetary gas(Takeuchi & Lin 2002; Lambrechts & Johansen 2012). Forsimplicity, we ignore the back-reaction of dust upon thegaseous nebular fluid for the time being, but we return to thisissue below (and quantify it in Appendix A). Under theassumption of a nearly Keplerian, circular orbit, the azimuthalequation of motion reads

» = --f f f

r

d r v

d t

v v

r

v v

t

1

2, 15r• • K •

fric

( )( )

where tfric is the frictional timescale relevant for the Epsteinregime of drag,

p rr

=ts

c8. 16fric

• •

s( )

6

The Astrophysical Journal, 894:143 (23pp), 2020 May 10 Batygin & Morbidelli

Note that tfric is approximately the isothermal sound crossingtime across particle radius, weighted by the solid-to-gasmaterial density ratio.

A key feature of Equation (15) is that the radial velocity ofthe particle vanishes in the limit where the particle azimuthalvelocity matches that of the gas. To examine if such a balanceis possible, let us consider the radial equation of motion:

= - W --fd v

d t

v

rr

v v

t. 17r r r• •

22 •

fric( )

The inertial term on the lhs can be set to zero, and followingTakeuchi & Lin (2002), we assume that both vf and fv • areclose to vK. Then, writing r Ω2 as h+fv v v rK K( ) , andretaining only leading-order terms, we have

h =-

--

=- --

f fv

rv

v v

r

v v

t

v

r

v t

r

v v

t

2 2

, 18

r r

r r r

K2

K• •

fric

K2

• fric •

fric( )

⎜ ⎟

⎜ ⎟

⎛⎝

⎞⎠

⎛⎝

⎞⎠

where we have employed Equation (15) to arrive at the secondline of the expression.

Setting =v 0r • , we trivially obtain an equilibrium solution for aparticle’s equilibrium frictional timescale, h=t r v v2rfric

eqK2( )( ) / . It

is further convenient to express this this quantity in terms of thedimensionless frictional time, i.e., the Stokes number t =eq( )

Wtfriceq( ) ,

th p h

= + =+S

v

v

M

M r21

1

4, 19req

Kmid

mid( ) ( ) ( )( )

◦

where the correction factor due to the midplane metallicity,9

+ 1 mid( ), originates from a marginally more detailed analysisthat accounts for the back-reaction of dust upon gas (seeAppendix A). For our adopted disk parameters and globalmetallicity that falls into the ~ 0.01 0.3– range, the aboveexpression evaluates to t ~ eq( ) - -10 105 3( – ).

The existence of a stationary =v 0r • solution toEquation (18) implies that as solid material enters thecircumplanetary nebula either through the meridional circula-tion or via ablation of planetesimals, dust with a physical radiusthat satisfies Equation (19) will get trapped in the disk. It isimportant to understand that qualitatively, this equilibriumstems from a balance between loss of particle angularmomentum due to aerodynamic drag and gain of angularmomentum due to coupling with a radial outflow of the gas.Consequently, this hydrodynamically facilitated process of dustaccumulation can only function in a decretion disk. Indeed, thesame process does not operate within circumstellar accretiondisks. This discrepancy brings to light an important distinctionbetween the formation of super-Earth-type extrasolar planetsand solar system satellites: despite having similar character-istics in terms of orbital periods and normalized masses, it islikely that their conglomeration histories are keenly distinct.

Examining the functional form of expression (19), we notethat because Σ is proportional to M , the equilibrium Stokesnumber is independent of the assumed mass-accretion rate.

Moreover, from Equations (7) and (12), it is easy to see thatt aµ

~req 3 4( ) , and thus, it varies by less than an order of

magnitude over the radius range of interest (i.e., ~rR0.1 0.3 H– ). Instead, this quantity is largely controlled by the

assumed value of the viscosity parameter and the midplanemetallicity (which is envisioned to slowly increase in time).For our fiducial value of α=10−4 and global metallicity10

of = 0.01, the typical equilibrium Stokes number is of theorder of t ~ -10eq 5( ) for the relevant disk radii. In terms ofphysical particle radius (for r = 1• g cm−3), this value trans-lates11 to ~ ´ -s few 10•

eq 2( ) cm. More precisely, s•eq( ) is

shown as a function of r in Figure 4. In addition to the linecorresponding to = 0.01, Figure 4 also depicts curvescorresponding to super-solar metallicities of = 0.1 and

= 0.3. The determination that s•eq( ) is always much larger

than a micron suggests that the dust-to-gas ratio of thecircumplanetary disk can increase dramatically without con-tributing an associated enhancement to opacity. Thus, theenvisioned picture is consistent with our assumption of anoptically thin disk.The assumption that interactions between solid particles and

gas lie in the Epstein regime is only justified as long asls 9 4•

eq( ) (where l s= n1 is the mean free path of gasmolecules). From Figure 4, it is clear that the equilibrium grainradius given by Equations (16) and (19) is smaller than λ

Figure 4. Equilibrium dust grain radius, s•eq( ), as a function of planetocentric

distance. Dust grains with radii bounded by the denoted range (i.e.,s•∼0.1–10 mm; Equation (19)) will remain trapped within the circumplane-tary disk, thanks to a balance between aerodynamic drag and radial updraft.Equilibrium curves corresponding to global disk metallicities of = 0.01, 0.1,and 0.3 (which translate to midplane metallicities of = 0.4, 14,mid and 74,respectively, see Equation (30)) are shown. The green line depicted on thefigure marks the transition between Epstein and Stokes regimes of drag,

l=s 9 4• t( ) , where λ denotes the mean free path of gas molecules. Thederived dust equilibrium is stable for constant particle size in the Epsteinregime of drag but not the Stokes regime. Although not directly modeled, it islikely that the dust growth/sublimation cycle can play an important auxiliaryrole in modulating s•. In particular, one can envision that because s•

eq( ) is adecreasing function of r in the Epstein regime, particle growth in the outer diskcan cause orbital decay. The reverse effect ensues at small orbital radii, wherethe sublimation of icy grains interior to the ice-line causes the particle size tofall below the equilibrium value, expelling solid material outward, where graingrowth can ensue once again, thereby maintaining ~s s• •

eq( ) on average.

9 The relationship between mid and is given by Equation (30).

10 As we demonstrate below, = 0.01 translates to 1mid , meaning thatthe correction factor + 1 mid( ) in Equation (19) is unimportant.11 Because t aµeq( ) , higher values of the Shakura–Sunyaev viscosityparameter would yield proportionally larger s•

eq( ). However, given thatgenerically, α0.01, the equilibrium dust radius for = 0.01 is unlikelyto exceed approximately a few centimeters.

7

The Astrophysical Journal, 894:143 (23pp), 2020 May 10 Batygin & Morbidelli

throughout most of the disk but not everywhere. This begs thequestion of what happens to solid material where the Epsteincriterion is not satisfied.

In a parameter regime where ls 9 4•eq( ) , we must

consider the Stokes regime of aerodynamic drag. For lowReynolds number flow (specifically, Re<1), the corresp-onding aerodynamic drag force is given by Weidenschilling(1977)

pr

pl

= = s vs v

c

12

ReRe

2. 20D •

2rel2 • rel

s( )

Conveniently, the corresponding frictional timescale—whichreplaces the expression given in Equation (16) whens•�9 λ/4—is independent of vrel, and has the form

pl

rr

= =

tm v s

c

2

9. 21fric

• rel

D

•2

•

s( )

In Figure 4, segments of equilibrium particle size curves thatcorrespond to Epstein and Stokes drag are shown by the blackand gray lines, respectively. Notably, the two regimes joinacross an equilibrium grain radius of l=s 9 4•

eq( ) , whereEquations (16) and (21) are equivalent. It is further worthnoting the change in sign of the derivative of s•

eq( ) with respectto r across this transition: in the Epstein regime, the equilibriumradius is a decreasing function of the planetocentric radius,whereas the opposite is true in the Stokes regime. As wediscuss below, this switch has important implications for thestability of the derived equilibrium.

Of course, this work is by no means the first to propose adust-trapping mechanism within quasi-Keplerian astrophysicaldisks. Rather, pressure maxima associated with long-livedvortices, boundaries between turbulent and laminar regions ofthe system, zonal flows, etc., have been widely discussed aspotential sites for localized enhancements of the nebular dust-to-gas ratio (see Varnière & Tagger 2006; Johansen et al. 2009;Lyra et al. 2009; Pinilla et al. 2012; Drążkowska &Szulágyi 2018 and the references therein). What differentiatesthe process outlined above from these ideas, however, is thefact that it stems from a balance between two large-scalefeatures of circumplanetary disk circulation and is thereforeglobal in nature.

If left unperturbed in a quiescent environment, the dustaccumulating within the circumplanetary disk would inevitablysink onto the midplane. The characteristic timescale on whichthis occurs is easily obtained from the Epstein drag equation(Armitage 2010) and is simply t= W 1settle

eq( )( ) . Diskturbulence, on the other hand, opposes dust settling bystochastically enhancing its vertical velocity dispersion. Giventhat the same turbulent eddies that drive vertical stirring alsofacilitate the viscous evolution of the disk, the diffusioncoefficient associated with turbulent stirring is often taken to bedirectly proportional to the disk viscosity parameter (Youdin &Lithwick 2007), n a= = W hSc Scturb

2 , where Sc is theturbulent Schmidt number.12

In the limit where the cumulative dust mass is negligible,competition among these two processes determines thethickness of the dust layer, and the expression for the solid

subdisk scale height has the form (Dubrulle et al. 1995)

t a=

+h

h

1 Sc. 22•

eq( )

( )

Note that in the regime where α = Sc τ(eq), we have h•=h,and Equation (22) simplifies to the oft-quoted result

a t»h h Sc•eq( )( ) . Our model circumplanetary disk, how-

ever, lies at the opposite extreme of parameter space. The dustlayer’s aspect ratio (h•/r) for our fiducial parameters, computedin the massless particle limit quoted above, is shown as a blackcurve in the top panel of Figure 3. Because α ? τ(eq), thevertical extent of the dust layer is comparable to that of the gasdisk everywhere in the nebula, meaning that purely hydro-dynamic settling of dust is exceedingly inefficient within thecontext of our model. Moreover, because t aµ S µ M1eq( ) is linearly proportional to the disk viscosity, this determinationis completely independent of the assumed value of α.A similar analysis can be undertaken for the radial diffusion

of aerodynamically trapped dust particles (Dullemond et al.2018). As a representative example, consider a particle thatsatisfies the equilibrium Equation (19) at the reference radius,r0. For our nominal parameters with + ~1 1mid( ) , thisparticle has a radius s•≈0.3 mm and lies in the Epsteinregime. Retaining this value of s•, the radial evolutionEquation (18) can be linearized around r=r0. Assuming that

p hS Wr M4 10 02 2( ) (which is very well satisfied for our

model), the linearized equation for the particle’s radial velocityin the vicinity of equilibrium takes on a rather rudimentaryform:

p» -

-S

vM r r

r4. 23r •

eq 0

0 02

( ) ( )( )

The fact that µ - -v r rr •eq

0( )( ) demonstrates that theequilibrium is stable to perturbations, since aerodynamic dragexerts a restoring force on the particle. Adopting the diskviscosity for the turbulent diffusion coefficient as before,13 anddefining the variable x x= - =r r v; r0 •

eq( ) , we obtain thefollowing stochastic equation of motion for the particle:

x ax

p= W -

Sd h d

M

rdt

4, 242

0 02

( )

W

where W represents a drift-free Weiner process(Øksendal 2013). An elementary result of stochastic calculusis that the solution to Equation (24) is a bounded random walk,with a characteristic dispersion

xp a

~S W

hr

M

4. 250 0

2

( )

The quantity inside the square root exceeds unity by a largemargin, implying that as long as the mass of the solidcomponent of the system is negligibly small, dust within thecircumplanetary disk is not only well-mixed vertically but mayalso experience relatively long-range radial diffusion. Inparticular, for nominal disk parameters, x ~ R0.17 H, implyingthat despite the functional form of the equilibrium (19),diffusion prevents size-sorting of particles within the disk.

12 The turbulent Schmidt number is a measure of turbulent viscosity relative tothe associated turbulent mixing.

13 For this problem, it is appropriate to drop the reduction factor Sc, since α isa viscosity parameter associated with radial angular momentum transport.

8

The Astrophysical Journal, 894:143 (23pp), 2020 May 10 Batygin & Morbidelli

Importantly, if we repeat this exercise in the Stokes regime(where tfric is given by Equation (21)) for a different nominalradius ¢r0, we find that p» - ¢ Sv M r r r7 8 .r •

eq0 0 0

2( ) ( )( ) Because in this case µ + - ¢v r rr •

eq0( )( ) , a particle perturbed

away from equilibrium will not experience a restoring radialacceleration and will instead flow away from ¢r0. This meansthat the derived equilibrium is only stable in the Epsteinregime. Although a useful starting point, this discussion ofstability along with Equation (24) should not be mistaken for aquantitative model of the radial distribution of particles,because the self-limiting cycle of grain growth and sublimationis likely to play an important dynamical role within theenvisioned circumplanetary nebula.

This can be understood as follows: as particles coagulate tolarger sizes, they experience stronger headwind (thus violatingthe equilibrium condition 19) and inspiral toward the planet.Upon crossing the ice-line of the circumplanetary disk,however, sublimation ensues, causing the particle size todiminish, until it is small enough for the dust to be expelledback out by the decretion flow. Beyond the ice-line, graingrowth ensues once again, and the cycle repeats. Therefore, weexpect that the particle distribution will be more stronglyconcentrated near the equilibrium radius than is suggested byEquation (24) if we were to account for dust sublimation andgrowth. Moreover, because the equilibrium particle radiusincreases with mid, for the successful operation of ourscenario, we must envision that the coagulation/sublimationcycle operates considerably faster than the growth of themidplane metallicity, thus maintaining ~s s•

eq( ) on averageeven as slowly increases.

4. Satellitesimal Formation

Our results from the previous section demonstrate that solidgrains of a particular size-range can be captured within thecircumplanetary nebula, by attaining a balance betweenaerodynamic drag and radial updraft associated with thedecretion flow. However, these particles have long settlingtimes and do not readily sediment into a vertically confinedsubdisk. Instead, as long as the overall metallicity, = S S • ,of the circumplanetary disk remains very low, we can envisionthat the gaseous and solid components of the system continueto be well-mixed. Nevertheless, it is clear that this simplepicture cannot persist indefinitely.

As the host planet continues to evolve within theprotoplanetary disk, incoming meridional circulation andablation of injected planetesimals act to slowly enrich thecircumplanetary disk in dust. This occurs at a rate

= +M f M M , 26• • • ablation( ) ( )

where f• is the mass fraction of incoming particles with Stokesnumbers corresponding to stationary values τ(eq) encapsulatedby the disk (and those that can grow to the appropriateequilibrium size before getting expelled from the disk).Although the precise value of M• depends on the adoptedsystem parameters (and is somewhat poorly constrained), it isworth noticing that a sufficient amount of solid mass to buildthe Jovian and Kronian moons (i.e., ~ ´ -M M 2 10•

4◦ ) can

be accumulated in a million years, even if planetesimal ablationis completely neglected and the mass fraction of incoming∼0.1–1mm particles is assumed to be a mere f•∼0.002—about

an order of magnitude smaller than the usual dust-to-gas ratio ofprotoplanetary disks (Armitage 2010).In light of the fact that in reality, M can be considerably

larger than our fiducial value of -M0.1 Myr 1◦ at earlier epochs

(Lambrechts et al. 2019), and that particle injection fromplanetesimal ablation can further increase the rate of dustbuildup within the system (Mosqueira et al. 2010; Ronnet &Johansen 2020), the aforementioned estimate almost certainlyrepresents a gross lower bound on the actual amount of solidmaterial that can be effectively sequestered in the disk.Moreover, as the parent (circumstellar) nebula gradually fades,the gaseous component of the circumplanetary disk must alsodiminish in time due to a decreasing M , thus graduallyenhancing . As a result, it seems imperative to consider thepossibility that a circumplanetary disk can readily approach astrongly super-solar metallicity (perhaps even of order unity)during its lifetime. Accordingly, let us examine the verticaldistribution of dust in a circumplanetary disk with 1.Independent of the degree of coupling that solid particles

experience with the gas, an inescapable limitation on thevertical dust stirring that turbulence can facilitate lies in theenergy budget of this process. That is, in order to lift the dustabove the midplane, turbulent eddies must do gravitationalwork, and the available kinetic energy to do this work isnecessarily restricted. Under the assumptions that underlieEquation (9), the characteristic velocity of turbulent eddies thatmanifest at the largest scales ( a~ℓ h) is of the order of

a~v cturb s. Thus, the column-integrated turbulence kineticenergy density is

ò ra

~ = S-¥

¥ v dz c

1

2 2. 27turb turb

2s2 ( )

When dust is lifted above the disk midplane, this kineticenergy gets converted into gravitational potential energy. In ageometrically thin disk, the downward gravitational accelera-tion experienced by dust can be approximated as g≈Ω2 z(Armitage 2010). Assuming that the solid component of thedisk follows a Gaussian profile like the gas, the column-integrated gravitational potential energy of the dust layer hasthe form:

ò r~ = S W-¥

¥ g z dz h . 28grav • •

2•2 ( )

Setting = turb grav, we arrive at the energy-limited expres-sion for the dust layer’s aspect ratio,

a~

h

r

h

r2, 29• ( )⎜ ⎟⎛

⎝⎞⎠

where ò � 1 is a numerical factor that accounts for the fact thatonly a fraction of the vertically integrated turbulence kineticenergy goes into elevating the dust above the midplane, as wellas for the suppression of turbulence by enhanced dustconcentration (Lin 2019). For definitiveness, in this work, weadopt ò=0.1 but remark that this guess is highly uncertain.The physical meaning of Equation (29) can be understood in

a straightforward manner: even if dust within the circumpla-netary disk is sufficiently well coupled to the gas for it topotentially remain well-mixed throughout the vertical extent ofthe disk, enhancing the scale height of the solid subdisk byturbulent stirring comes at a steep energetic cost. As a result, in

9

The Astrophysical Journal, 894:143 (23pp), 2020 May 10 Batygin & Morbidelli

relatively quiescent, dust-rich systems (where a ), thesolid layer will necessarily be thin compared to the gas disk.14

Proceeding under the assumption of a Gaussian profile asbefore, we may readily write down the functional form of themidplane metallicity by taking the ratio of midplane densities:

a=

SS

~

h

h

2. 30mid

•

•( )

Noting that α ò∼10−5, we remark that in order for mid toexceed unity, the overall disk metallicity must only exceed thesolar value by a factor of a few. The ensuing verticalconfinement of dust has profound consequences for theformation of satellitesimals.

By now, it is well-established that a broad range of gravito-hydrodynamic instabilities can develop within two-fluidmixtures of gas and dust (Youdin & Goodman 2005; Johansenet al. 2007; Squire & Hopkins 2018; Seligman et al. 2019).These remarkable phenomena, however, only emerge at asufficiently high (local) concentration of high-metallicitymaterial. A broadly discussed example of this group ofinstabilities is known as the streaming instability (Youdin &Goodman 2005; Johansen & Youdin 2007), which canfacilitate rapid growth of dust clouds within protoplanetarydisks through a back-reaction of accumulated solid particles onthe background quasi-Keplerian flow. A distinct variant of atwo-fluid instability is known as the > 1 resonant draginstability (Squire & Hopkins 2018) and can also promote thecoagulation of solid material within the disk, albeit at smallerscales. Importantly, within the context of the protosolar nebula,it is now widely speculated that enhancement of the local solid-to-gas ratio associated with the aforementioned effects canculminate in the gravitational collapse of particle clouds,resulting in the formation of bona fide planetesimals.

The emergence of gravito-hydrodynamic resonant draginstabilities within dust-loaded circumplanetary decretion disksis an intriguing possibility that deserves careful investigationwith the aid of high-resolution numerical simulations. At thesame time, this exercise falls beyond the immediate scope ofour (largely analytic) study. Accordingly, to circumvent thisriveting complication, here, we focus our attention on thequalitatively simplest pathway for conversion of dust intosatellitesimals: sedimentation, followed by direct gravitationalcollapse, i.e., the Goldreich–Ward mechanism (Goldreich &Ward 1973; see also Youdin & Shu 2002).

Linear stability analysis of differentially rotating self-gravitating disks carried out over half a century ago(Safronov 1960; Toomre 1964) has shown that in order forgravitational collapse to ensue in the presence of Keplerianshear, the system must satisfy the following rudimentarycriterion:

p pa

=WS

=WS

Q

h h

21. 31•

•2

•

2

3( )

For our adopted benchmark parameters of = -M M0.1 Myr 1◦

and α=10−4, this expression dictates that direct conversion ofdust into satellitesimals can be triggered within the circumpla-netary disk for 0.2 (Figure 5). Equations (12) indicate that

µ~

-Q r•3 4, implying that, like in the case of circumstellar

nebulae (see, e.g., Boss 1997), gravitational collapse is moreeasily activated in the outer regions of our model circumpla-netary disk. Notably, this preference for longer orbital periodsfor the generation of satellitesimals is generic and would applyeven if pre-collapse agglomeration of solids was assisted bysome two-fluid instability (Yang et al. 2017 and the referencestherein).It is well known that our envisioned process for satellitesimal

formation (where dust consolidates directly into satellitesimalsunder its own gravity) can be suppressed under certainconditions in real astrophysical disks. To this end, Goldreichet al. (2004) point out that in order for gravitationalfragmentation to ensue, the particle disk must be opticallythick. Quantitatively, this criterion translates to rS s 1• • •( ) .This limit does not pose an issue for the problem at hand,because the smallness of the equilibrium particle size withinour disk (as dictated by Equation (19)) ensures that thisinequality trivially satisfied.A more acute suppression mechanism for the Goldreich–

Ward instability is the turbulent self-regulation of dust settling.That is, as the dust layer is envisioned to grow thinner, itsmidplane azimuthal velocity inevitably approaches the purelyKeplerian value. The shear associated with the development ofa vertical gradient in vf gives rise to the Kelvin–Helmholtzinstability, which turbulently stirs the dust layer, counteractingsedimentation (Weidenschilling 1980; Cuzzi et al. 1993).While this process can indeed subdue planetesimal formationin the circumstellar nebula where ~ 0.01, for the circum-planetary system at hand, this problem is circumvented byvirtue of the disk having a sufficiently high metallicity. In otherwords, even if the solid subdisk is perturbed by turbulence, thedust cannot be lifted appreciably due to energetic limitations(Equation (29)). This reasoning is supported by the results ofSekiya (1998), who demonstrated that for 0.1, duststirring becomes inefficient, allowing gravitational collapse toproceed even in the presence of parasitic Kelvin–Helmholtzinstabilities. Consequently, we conclude that while thedevelopment of Kelvin–Helmholtz instabilities can suppress

Figure 5. Gravitational stability of the solid subdisk. Toomre’s Q parameter(Equation (31)) is shown as a function of orbital radius, for a sequence ofcircumplanetary nebula metallicities. For solar composition gas with = 0.01,Q•?1, and the solid subdisk is gravitationally stable. However, for a dust-to-gas ratio in excess of 0.2, the solid subdisk becomes gravitationallyunstable, fragmenting into satellitesimals. Notably, for = 0.3—which wetake as a reasonable estimate for the onset of large-scale satellitesimalformation—gravitational collapse can ensue outwards of r R0.1 H.

14 This effect highlights yet another important distinction between the physicsof satellite formation and planet formation. In typical protoplanetary disks,

~ 0.01, and dust sedimentation toward the midplane occurs simply due tothe fact that for a broad range of particle sizes, α = τ(eq) (see Equation (22)).

10

The Astrophysical Journal, 894:143 (23pp), 2020 May 10 Batygin & Morbidelli

the Goldreich–Ward mechanism in the protosolar nebula, thedirect gravitational collapse of solid grains into planetesimals ispossible in circumplanetary disks because of dust-loadingwithin the system.

The dispersion relation associated with a self-gravitatingKeplerian particle fluid has the following well-known form(Armitage 2010):

w p= - S + Wc k k2 , 322•2 2

•2∣ ∣ ( )

where =c h r v• • K( ) is the velocity dispersion of the dust. Thecritical wavenumber corresponding to the most rapidly growingunstable mode of this relation is

p=

S~

k

c h

1, 33crit

•

•2

•( )

where the rhs follows from setting Q•∼1 in Equation (31).Accordingly, the characteristic mass scale of planetesimalsgenerated through gravitational collapse is

pp

p a~ S = Smk

h2

2 . 34crit

2

•3 2 ( )

⎛⎝⎜

⎞⎠⎟

Importantly, to arrive at the rhs of this estimate, we have usedthe expression in Equation (29) for h• to cancel out thedependence on .

The fact that this expression is independent of isqualitatively important. Indeed, while the metallicity dictateswhether or not gravitational collapse can be triggered viaEquation (31), the physical properties of satellitesimalsgenerated through fragmentation of the solid subdisk arelargely determined by the global properties of the circumpla-netary nebula. For our fiducial parameters, we obtain bodieswith m∼1019 kg at ~r R0.1 0.3 H– —comparable to the massof Mimas and about two orders of magnitude smaller than themass of Iapetus. For a mean density of 1 g cm−3, this massscale translates to bodies with radii of the order of 100 km.

5. Satellite Growth and Migration

While the characteristic mass scale of satellitesimalsgenerated through gravitational instability is appreciable, it isstill negligible compared to the cumulative mass of the Galileanmoons or Titan, meaning that additional growth must takeplace to explain the satellite systems of the giant planets.Growth of solid bodies within the circumplanetary disk canproceed via two potential pathways: pair-wise satellitesimalcollisions or pebble accretion. Recently, the pebble accretionparadigm has been shown to be remarkably successful inresolving long-standing issues of planet formation (Ormel &Klahr 2010; Lambrechts & Johansen 2012; Morbidelli &Nesvorny 2012; Ronnet & Johansen 2020) and has conse-quently gained considerable traction within the broadercommunity. Inspired by this mechanism’s growing fashion-ableness, let us begin this section by considering the growth ofsatellitesimals by pebble capture within the circumplane-tary disk.

5.1. Pebble Accretion

Depending on the mass of the growing satellitesimal and thedegree of dust–gas coupling, its propensity toward aerodyna-mically assisted capture of small particles can proceed in one of

two modes of accretion: the Bondi regime or (the considerablymore efficient) Hill regime. In the former case, the key physicallength and timescales that characterize the pebble accretionprocess are the Bondi radius, and the corresponding crossingtime

=D

=D

R

vt

R

v, 35B 2 B

B ( )

where is the satellite embryo’s mass, and hD =v vK

+ 1 mid( ) is the particle approach speed. Qualitatively, RB

represents a critical impact parameter below which the gravita-tional potential of the growing body can facilitate large-angledeflection of dust, and tB is the characteristic timespan associatedwith the encounter.Contrary to the case of the protosolar nebula, where a

broad size distribution of dust particles translates to anextended range of Stokes numbers, the aerodynamic equili-brium delineated in Section 3 ensures that the circumplanetarydisk is loaded with solid particles that are characterized by asimilar frictional timescale, tfric

eq( ). This allows us to define analmost unique capture radius for circumplanetary dust in theBondi regime (Ormel & Klahr 2010),

h» =

+

R R

t

t v

r v v1

2, 36r

cB

Bfric

B

mid

K2

K( ) ( )

where we have used Equation (19) to relate tfric to vr.In the Hill regime, the effective capture radius is (Ida et al.

2016)

th

» =+

R rM

rv

v M

10

3

5 1

3. 37r

cH mid

K

13

13( ) ( )

◦ ◦

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

The crossover between the two modes of accretion occurswhen the effective Bondi and Hill accretion radii areequivalent, i.e., ~R Rc

BcH. After some rearrangement (see

Appendix B), this yields a transitionary embryo mass of theorder of

p h~

+S W

r

MM

400

9 1. 38t

mid

4 2( )◦

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

If we assume that the conversion of dust into solid bodies isless than 100% efficient, such that following large-scalesatellitesimal formation, the pebble surface density is still onthe order of Σ•/Σ∼0.1 (which corresponds to » 14mid ),the above expression suggests that the Hill regime of accretionis appropriate for satellites more massive than ~ t

´few 1024 kg. This mass scale exceeds the mass of Ganymedeby more than an order of magnitude, implying that anyaccretion of pebbles in our model circumplanetary disk is sureto proceed in the Bondi regime.For 1022 kg, R hc

B•. Recalling that the characteristic

mass scale of satellitesimals that form via gravitational collapseis m∼1019 kg, this means that the dust layer is much morevertically extensive than the pebble capture radius, implyingthat accretion unfolds in 3D. The corresponding accretion rate

11

The Astrophysical Journal, 894:143 (23pp), 2020 May 10 Batygin & Morbidelli

has the form (see Appendix B for additional details)

p

hpa

=S

D

=+

S-

d

d t hR v

M

h

rr v

2

1

2

1, 39r

B 3D

•

•cB 2

mid1 3

( )

( )◦

⎜ ⎟

⎜ ⎟

⎛⎝

⎞⎠

⎛⎝

⎞⎠

where we have adopted pS h2• •( ) as an estimate for thevolumetric density of the pebble disk. Setting ~ 0.1, thisformula yields a mass-doubling time of ´ ~- d d t1 1( )2000 yr.

While the above estimate of relatively rapid accretion mayappear promising, it is unlikely that it translates to significantlong-term growth. This is because within the context of ourmodel, pebble accretion is necessarily limited by the satellite-simal’s access to the overall supply of dust. That is, unlike theoft-considered case of a circumstellar accretion disk, whereinward drift of pebbles acts to refill the orbital neighborhood ofgrowing planetesimals, in the circumplanetary disk, theaerodynamic equilibrium discussed in Section 3 implies thatno steady-state drift exists. Instead, here, the radial dispersionof pebbles is driven almost entirely by turbulence viscosity,yielding diffusion-limited growth of satellitesimals that isreminiscent of planetesimal growth within dust-loaded pressurebumps (recently considered by Morbidelli 2020). Thus, crudelyspeaking, the reservoir of solid dust that is available to anygiven satellitesimal is restricted to the material that is entrainedbetween the satellitesimal itself and its nearest neighbors.

If we envision that large-scale gravitational collapse of thesolid subdisk yields a population of debris that is comparable intotal mass to that of the remaining dust disk, and that thegenerated satellitesimals commence their growth at approxi-mately the same time, the above reasoning implies that thepebble accretion process can only boost the individual massesof satellitesimals by a factor of ∼2 before the global supply ofdust is exhausted. We therefore conclude that within thecontext of our model, pebble accretion can only yield a short-lived burst of satellitesimal growth and is unlikely to be thedominant mechanism for converting satellitesimals into full-fledged satellites. In light of the short-lived nature of thisprocess, coupled with considerable uncertainties on theefficiency its operation, we will neglect it for the remainderof the paper.

5.2. Oligarchic Growth

If pebble accretion is ineffective in boosting the masses ofsatellitesimals by an appreciable amount, long-term conglom-eration of the large giant planet satellites must occur throughpair-wise collisions.15 In this case, the rate of accretionexperienced by a satellite embryo is dictated by an n–σ–v-type relation and has the form (Lissauer 1993)

p r k p= = L S + Q W

d

d t

d

d t4 1 , 402 2¯ ( ) ( )

where Λ�1 is the efficiency of the conversion of dust intosatellitesimals through gravitational instability, κ is a constantof order unity,16 and Q = á ñv vesc

2( ) is the Safronov number.

Qualitatively, the parameter Λ regulates the total mass of thesatellitesimal swarm in the region where Q•1. Given thatneither the satellitesimal generation process nor the satelliteaccretion process are expected to be perfectly efficient, a valueof Λ∼1/3–1/2 appears reasonable within the framework ofour model. We note, however, that any value of Λ in excess of∼0.1 yields a debris disk between R0.1 H and R0.3 H thatexceeds the total mass of the observed satellites.The solution to Equation (40) for as a function of t is

trivially obtained if the mean density and the parameters on therhs of the differential equation are assumed to be time-invariant. For the purposes of the following discussion, it isinstructive to recast this solution in terms of an accretiontimescale, accr, corresponding to a change in the embryo’sradius from 0 :

rk

=-

L + Q S W

4

1. 41accr

0¯ ( )( )

( )

By neglecting gravitational focusing in the above expression(setting Q 0), we can obtain an approximate upper limit onthe formation timescale of large satellites, accr

max , as a functionof r in our model disk. Retaining the same system parametersas those delineated in the proceeding sections, and setting

= 2600 km and r = 2¯ g cm−3 (approximate radius andmean density of Ganymede, respectively), we show accr

max inFigure 6 for Λ=1, 1/2, and 1/4. Crucially, this resultdemonstrates that the characteristic conglomeration timescalecan reasonably exceed a million years, provided that thevelocity dispersion of satellitesimals exceeds the escapevelocity of the satellite embryo.In the opposite limit, where the satellitesimal swarm is taken

to be (initially) dynamically cold, embryo growth can proceedon a much shorter timescale, at first. However, the satellite-simal swarm cannot remain dynamically cold indefinitely, dueto self-stirring and interactions with the accreting embryo—acaveat best addressed with the aid of detailed simulations. Inany case, the accretion process necessarily stops once thenewly formed satellite is ejected from the annulus of the

Figure 6. Maximal accretion timescale of satellites (Equation (41)). Thedepicted curves correspond to a disk metallicity of = 0.3 and a satellitesimalgeneration efficiency of 100% (Λ=1), 50% (Λ=0.5), and 25% (Λ=0.25).The mean density and terminal radius are taken to be r = 2¯ g cm−3 and

= 2600 km, respectively. While we assume the Safronov number, Θ, to benull for the purposes of this figure, it is important to keep in mind thatgravitational focusing can significantly accelerate satellite formation, if thesatellitesimal velocity dispersion is low. Note further that the accretion rate issemimajor-axis-dependent and proceeds more than an order of magnitude fasterat ~r R0.1 H than at ~r R0.3 H.

15 Notably, this mode of accretion is qualitatively much closer to the standardpicture of terrestrial planet formation than it is to the emergent picture of theformation of super-Earths.16 For an isotropic velocity dispersion, k = 3 2 (Lissauer 1993).

12

The Astrophysical Journal, 894:143 (23pp), 2020 May 10 Batygin & Morbidelli

circumplanetary disk occupied by the satellitesimal swarm.Within the context of our scenario, we envision this to occur asas consequence of satellite–disk interactions (Goldreich &Tremaine 1980; Ward 1997), which sap the satellite of itsorbital angular momentum, leading to its progressively rapidorbital inspiral. The characteristic timescale for the satellite’sinward (type-I) migration is given by Tanaka et al. (2002),

g=

W S

M M

r

h

r, 42mig 2

2

( )◦ ◦ ⎜ ⎟⎛⎝

⎞⎠

where γ is yet another dimensionless constant of order unity.Importantly, the expression in Equation (42) states that the

migration timescale is inversely proportional to the satellitemass. This means that long-range orbital decay cannotensue until the satellite is sufficiently large. Consequently, toobtain a crude limit on (or), we follow Canup & Ward(2002, 2006) and set ~ mig accr. After some rearrangement,we have

k gp r

- ~ L + Q h M

r

3

161 . 433

0 2

2

( )¯

( ) ( )◦⎛⎝⎜

⎞⎠⎟

While a closed-form expression for does exist, it iscumbersome and does not elucidate any physics that is notalready evident upon inspection of Equation (43). Accordingly,here we limit ourselves to simply noting that the solution foris roughly given by the fourth root of the rhs of the aboveexpression and that this approximation improves for larger (recall that0 is set by the typical satellitesimal mass, given byEquation (34)).

The above discussion indicates that the terminal radius of asatellite is determined by two parameters: the dynamicaltemperature of the satellitesimal swarm Λ(1+Θ) and theplanetocentric radius r. Contours of terminal are shown inFigure 7, with real satellite radii marked by the colored lines.Remarkably, this rudimentary analysis suggests that satelliteswith radii in the range ~ 1500 2500– km can be naturallygenerated within the circumplanetary decretion disk, providedthat satellitesimal disk that forms them originates with a lowvelocity dispersion.

6. Numerical Experiments

Without a doubt, the actual process of satellite formation ismore complicated than the narrative foretold by the simplecalculations presented above. Accordingly, it is imperative thatwe examine the validity of the emerging picture with moredetailed numerical simulations. This is the primary purpose ofthis section.

6.1. Accretion Calculation

In order to test the growth of a massive satellite embryo in thesatellitesimal disk described in Section 5.2, we have used aparticle-in-a-box code, Boulder, developed and described inMorbidelli et al. (2009). The code accounts for the self-stirring ofeccentricities and inclinations of the satellitesimal disk, as well ascollisional damping, gas drag, and dynamical friction. The latterdamps the eccentricity and inclination of the most massive objectsat the expense of causing the smallest particles to becomedynamically excited. Collisions between particles were treatedaccording to the prescription of Benz & Asphaug (1999) such that

they could result in perfect merging, partial accretion, erosion, orcatastrophic break-up, depending on collision velocities and sizesof the impacting bodies.Our initial conditions represent an annulus of debris centered

at =r R0.2 H with a full width ofD =r R0.1 H, i.e., the middleof the range illustrated in Figures 6 and 7, spanning0.15– R0.25 H. Initial satellitesimals were assigned a mass ofm=1019 kg and a radius of 100 km in agreement with theestimate of Section 4. The total mass of the satellitesimalpopulation in the annulus was taken to be Mdisk=3.2×1023 kg. This corresponds to a gas density of Σ=1000 g cm−2

at =r R0.2 H, in approximate agreement with our nominalprofile of Equation (7), a solid-to-gas ratio of = 0.3, and asatellitesimal formation efficiency of 30%, uniformly spreadover the annulus (the same parameters have been used inFigure 7). Initially, the eccentricities and inclinations were setto á ñ » ´ -e 6 10 6 and á ñ » ´ -i 1.5 10 deg5 , respectively.Importantly, however, these quantities evolved rapidly, suchthat after only 100 yr, the eccentricity and inclination 100 kmsatellitesimals were already á ñ =e 0.014 and á ñ =i 0.4 deg,respectively, growing further to á ñ =e 0.04 and á ñ =i 1 deg bythe t=1000 yr mark.The key advantage of a code like Boulder over the analytic

calculations presented in Section 5.2 is that the code computesthe gravitational focusing factor self-consistently, from themasses of the colliding bodies and their mutual velocity. Thatis, because of dynamical excitation within the disk, for a giventarget, the gravitational focusing factor decreases over time.However, because the most massive bodies grow more readily

Figure 7. Terminal radii of satellites. Within the framework of our model,satellite conglomeration continues until long-range orbital decay ensues andremoves the growing embryo from its r R0.1 H feeding zone. Accordingly,the terminal mass, and radius, of a forming satellite are approximatelydetermined by equating the (type-I) migration timescale to the accretiontimescale (Equation (43)). Contours of terminal satellite radii equal to 1000,1500, 2000, and 2500 km are shown by the black curves in the figure. Contourscorresponding to the true radii of the Galilean satellites and Titan are depictedwith colored lines and are labeled. A disk metallicity of = 0.3 and a meansatellite density of r = 2¯ g cm−3 are assumed. The precise values of order-unity constants κ and γ are ignored.

13

The Astrophysical Journal, 894:143 (23pp), 2020 May 10 Batygin & Morbidelli

(Safronov 1969), their focusing factor can instead increase,provided their escape velocity increases faster than the velocitydispersion in the disk.

Panel (A) of Figure 8 depicts the growth of the largest objectwithin the annulus as a function of time. A radius of

= 2600 km is reached in a bit less than 100,000 yr—aboutan order of magnitude faster than estimated in Figure 6, wheregravitational focusing is neglected. We also note that embryogrowth is more complex than that envisioned in Section 5.2(i.e., accretion at a constant rate), as mergers with other massivebodies cause the satellite embryo’s radius to sporadically jumpupwards. Indeed, this is typical of the oligarchic growth process(Kokubo & Ida 1998).

Panel (B) of Figure 8 shows the cumulative size–frequencydistribution of the satellitesimal population after 84,000 yr. Inaddition to the aforementioned = 2600 km body, there is asecond body slightly exceeding 1000 km in radius. Furtherdown the radius ladder, there is one body with ~ 700 kmand seven with ~ 500 km. In summary, only a handful ofmassive satellite embryos emerge form the annulus, with thevast majority of objects remaining small or even decreasing insize because of collisional fragmentation.

6.2. N-body Simulations

The above simulation demonstrates that the planetesimalsubdisk generated by gravitational collapse of dust is conduciveto the emergence of isolated massive embryos. This particle-in-a-box calculation, however, cannot capture the global dynamicsof the system, which must instead be modeled with the aid ofdirect N-body simulations. Accordingly, we have carried out aseries of numerical experiments that track the long-term orbitalevolution of growing embryos, subject to gravitationalcoupling as well as disk–satellite interactions.

The initial conditions adopted in our N-body experimentsdraw upon the results of Sections 3, 4 and 6.1. In particular, oursimulations began with a dynamically cold (á ñ ~ á ñ ~ -e i 10 3) seaof satellitesimals, extending from =r R0.1in H to =r R0.3out H.The effective disk surface density followed a∝r−5/4 profile

as dictated by Equation (12) but with a diminished value of Σ0.Keeping in mind that accretion is not expected to be 100%efficient, the total mass of the planetesimal swarm was chosen to be

= ´ -M M6 10disk4

◦, i.e., approximately three times the totalmass of the Galilean satellites. We note that in terms of our modelcircumplanetary nebula outlined in Section 2, this planetesimal diskis about an order of magnitude less massive than the cumulativegas mass contained in the same orbital region and, as before,effectively translates to ~ L ~ 0.3.To save computational costs, the planetesimal swarm was

modeled as 1000 semiactive m≈1021 kg superparticles. Thesesuperparticles were allowed to gravitationally interact with thecentral planet and the satellite embryos but not among themselves.Each supersatellitesimal was also subjected to aerodynamic dragensuing from the circumplanetary nebula, employing the accel-eration formulae of Adachi et al. (1976). Despite being two ordersof magnitude more massive than satellitesimals that are envisionedto result from gravitational fragmentation of the solid subdisk, theaerodynamic drag calculation was carried out treating the particlesas = 100 km, r = 1¯ g cm−3 bodies.The simulations were initialized with three satellite seeds

(a separate discussion of the formation of Callisto will bepresented below) with negligible masses, placed randomlybetween R0.1 H and R0.3 H. Collisions between these proto-satellites and satellitesimals were treated as perfect mergers. Inaddition to conventional N-body interactions with the centralplanet and the planetesimal swarm, the satellite embryosexperienced both aerodynamic drag (computed self-consis-tently, assuming r = 1¯ g cm−3), as well as type-I migration andorbital damping, which were implemented using the formulaeof Papaloizou & Larwood (2000). The migration andeccentricity/inclination damping timescales were taken to bemig (Equation (42)) and = = - h r 10damp

2mig

2mig( ) ,

respectively.The calculations were carried out using the mercury6

gravitational dynamics software package (Chambers 1999). Thehybrid Wisdom–Holman/Bulirsch–Stoer algorithm (Wisdom &Holman 1991; Press et al. 1992) was used throughout, with a

Figure 8. Particle-in-a-box calculation of satellite accretion within the circumplanetary disk. Panel (A) depicts a time-series of the largest satellite embryo’s physicalradius. Facilitated by efficient gravitational focusing, the embryo experiences rapid initial growth. However, the rate of accretion slows down in time, as the velocitydispersion of the satellitesimal disk becomes progressively more excited. Intermittent jumps in radius correspond to collisions of the protosatellite with other massiveembryos within the system; 84,000 yr into the simulation time, the protosatellite attains a radius of 2600 km—comparable to that of Ganymede. Panel (B) shows thecumulative size–frequency distribution of the system at t=84,000 yr. Importantly, this panel demonstrates that the aftermath of accretion within the circumplanetarydisk is highly uneven, such that only two protosatellites larger than 1000 km emerge at the end of the simulation. In addition to these two bodies, a single

» 700 km object, along with seven » 500 km embryos occupy the R0.15 0.25 H– satellitesimal annulus. On the smaller end of the size–frequency distribution, aprolonged tail of collisionally generated debris extends below 0.1 km. Cumulatively, this calculation suggests that conditions within the circumplanetary disk arepropitious to the emergence of a small number of massive embryos.

14

The Astrophysical Journal, 894:143 (23pp), 2020 May 10 Batygin & Morbidelli

time-step of Δt=1 day and an accuracy parameter of = - 10 8ˆ .Any objects that attained a radial distance in excess of a JovianHill radius were removed from the simulation. Additionally, anyobjects that attained an orbital radius smaller than R0.03 H(roughly the present-day semimajor axis of Ganymede) wereabsorbed into the central body. This was done to maintain areasonably long time-step, with the understanding that the processof capturing Io, Europa, and Ganymede into the Laplaceresonance would have to be simulated a posteriori.

We ran 12 such numerical experiments in total, eachspanning 0.1Myr. Qualitatively, simulation results followedthe expectations of the analytical theory outlined in theprevious section. That is, growth of typical satellite embryoswas terminated primarily by their departure from the debris

disk through inward type-I migration. Moreover, the satelliteconglomeration process—once complete—would leave behinda dynamically excited sea of satellitesimals, preventing the nextsatellite from forming until the system would recircularize byaerodynamic drag.Figure 9 shows a series of snapshots of one particularly

successful run, spanning ∼1–30 kyr. This specific simulationyields three satellites that bear a striking resemblance to Io,Europa, and Ganymede both in terms of mass as well as orbitalordering. Within the context of this numerical experiment,owing to an initially low velocity dispersion among satellite-simals, the first (closest in) satellite seed experiences rapidgrowth (panel (A)). In only ∼2 kyr, the satellite attains a massequal to 101% of Io and sets off on an extended course of

Figure 9. Formation of the three inner Galilean satellites. An initially dynamically cold (e∼i∼10−3) disk of 1000 supersatellitesimals, comprising= ´ -M M6 10disk

4◦ is assumed to form by gravitational fragmentation between =r R0.1in H and =r R0.3out H (see Section 4). Three satellite seeds are introduced

within the same orbital range, at random planetocentric distances. In terms of disk metallicity, satellitesimal formation efficiency and the Safronov number, these initialconditions translate to ~ L ~ 1 3 and Θ∼400. Satellite seeds destined to become Io, Europa, and Ganymede are depicted in gray, blue, and purple, respectively,and their sizes serve as a proxy for their physical radii. On the other hand, colors of semiactive superparticles inform their orbital inclinations, as shown in the leftcolumn. In addition to gravitational dynamics, the effects of aerodynamic drag and gas disk-driven migration are self-consistently modeled in this simulation. Resultsof this numerical experiment are summarized as follows. Owing to gravitational focusing in an initially pristine disk, the conglomeration of Io begins quickly andunfolds on a relatively short (∼1000 yr) timescale (panel (A)). Then, 2500 yr into the simulation, Io decouples from the satellitesimal feeding zone and begins tomigrate toward the Jovian magnetospheric cavity (panel (B)). As Io’s orbit decays, Europa’s growth ensues (panel (C)). However, due to an already-excited velocitydispersion among satellitesimals, Europa’s accretion is somewhat less efficient, and by the ∼10,000 yr mark, Europa detaches from the satellitesimal disk, havingachieved a smaller terminal mass than Io (panel (D)). For the following ∼104 yr, aerodynamic drag acts to re-cicularize the satellitesimal disk (panel (E)), and theconglomeration process restarts approximately ∼25,000 yr into the simulation (panel (F)). Ganymede achieves its terminal mass shortly thereafter (panel (G)), and by30,000 yr, it follows Io and Europa on an inward migratory trek (panel (H)).

15

The Astrophysical Journal, 894:143 (23pp), 2020 May 10 Batygin & Morbidelli

orbital decay (panel (B)). Meanwhile, a second embryo beginsits conglomeration process (panel (C)). However, due to a pre-excited orbital distribution of satellitesimals, this embryo growsmore slowly and leaves the satellitesimal disk at the ∼10 kyrmark, having attained a lower mass, equal to 97% of that ofEuropa (panel (D)).

For the ∼10 kyr that follow, a large orbital eccentricity andinclination of the outermost satellite seed are maintained by thedynamically hot satellitesimal swarm (panel (E)). However,aerodynamic drag eventually recircularizes the debris, andrunaway growth of the final seed ensues approximately 25 kyrinto the simulation (panel (F)). As with the first embryo,conglomeration proceeds rapidly, and the final seed reaches amass equal to 104% of that of Ganymede in only a fewthousand years (panel (G)). By the 30 kyr mark, the thirdsatellite leaves the satellitesimal feeding zone and sets off on asteady path of inward migration.

While this particular simulation provides the best match to theactual Galilean satellite masses, it is not anomalous within thebroader context of our simulation suite. In particular, almost all ofour runs generated satellites with masses that are comparable(within a factor of∼3) to that of Io, and four out of 12 simulationsended with the correct mass ordering, wherein the least massivesatellite is generated in between two more massive ones.Figure 10 shows the outcome of our complete simulation suitewhere satellite mass is plotted as a function of the orbital radius.The results of the particular simulation depicted in Figure 9 arehighlighted with thick lines. As an additional check on ourcalculations, we have carried out similar simulations using thesymba integrator package (Levison & Duncan 2000) employinga marginally different implementation of aerodynamic drag andtype-I migration, as well as a different N-body algorithm, and weobtained similar results.

As a corollary, we remark that in some of our simulations, aninward-migrating Io captured a few satellitesimals into interiorresonances, shepherding them onto very short-period orbitsaround Jupiter. Within the context of our model, we mayenvision that after the dissipation of the circumplanetary disk, atidally receding Io would break resonance with these bodies,leaving them to encircle Jupiter to this day. Such a picture isremarkably consistent with the existence of the Amalthea group

of Jovian satellites—a collection of four ~ 10 100– kmobjects possessing P∼7–16 hr orbital periods.

6.3. Formation of the Laplace Resonance

Among the most iconic and well-known characteristics ofthe three inner Galilean satellites is their multiresonant orbitalarchitecture. While an understanding of the celestial machineryof this resonance dates back to the work of Laplace himself, thedynamical origin of the 4:2:1 commensurability was onlyelucidated a little over half a century ago. In particular,Goldreich (1965) was the first to propose that slow outwardmigration, facilitated by tidal dissipation within Jupiter,provides a natural avenue for the sequential establishment ofa multiresonant lock among the inner satellites. In the decadesthat followed, the plausibility of the tidal origin hypothesis wasfurther corroborated with increasingly sophisticated numericalmodels (Peale 1976; Henrard 1982; Lari et al. 2020).An alternative picture—proposed by Peale & Lee (2002)—is

that although tidal dissipation is undoubtedly an active process,the 4:2:1 orbital clockwork connecting Io, Europa, andGanymede is primordial. More specifically, Peale & Lee(2002; see also Canup & Ward 2002) suggest that the Laplaceresonance was established before the dissipation of the circum-Jovian nebula as a result of convergent inward migration,driven by disk–satellite interactions. Because both the tidalmigration and disk-driven migration scenarios can, in principle,reproduce the current orbital architecture of the satellites, it isdifficult to definitively differentiate between them. Never-theless, it is obvious that disk-driven assembly of the Laplaceresonance ensues naturally within the context of our model, andto complete the qualitative narrative proposed herein, weexplore this process numerically.Recall that the N-body simulations carried out in the previous

subsection (and illustrated in Figure 9) point to sequential satelliteformation, where upon accruing a sufficiently large mass, agrowing object exits the satellitesimal disk via inward type-Imigration. If the gaseous component of the circumplanetary diskwere to extend down to the planetary surface, the inspiralingsatellite would simply be engulfed by the planet. However, asalready mentioned in Section 2, rudimentary considerations of therelationship between magnetic field generation and giant planetluminosity during final stages of accretion suggest that thecircumplanetary disk is likely to be truncated by the planetarymagnetosphere at a radius of ~R R5T Jup (Batygin 2018;Ginzburg & Chiang 2020). Accordingly, the inner edge of thenebula should act as a trap that halts the orbital decay of the firstlarge satellite (Io) at »r RT.We begin our simulations of Laplace resonance assembly at

this stage. Io is assumed to start at its current orbital location (avalue approximately equal to RT), while Europa and Ganymedeare initialized out of resonance, with semimajor axes factors oftwo and four greater than that of Io, respectively. Rather thanattempting to emulate the effects of the satellite trap on Iothrough sophisticated parameterization of type-I migration (see,e.g., Izidoro et al. 2019), we opt for a simpler procedurewherein the semimajor axes of all satellites in the calculationare renormalized at every time-step17 such that the orbitalperiod of the innermost body is always equal to that of Io (see,

Figure 10. Satellite formation tracks obtained within our full simulation suite,shown on a mass–orbital radius diagram. Generally, the terminal mass ofobjects generated in our N-body experiments is similar to that of the realGalilean satellites. Furthermore, the correct mass ordering of the bodies isreproduced in four out of 12 instances. The specific formation tracks of Io,Europa, and Callisto shown in Figure 9 are highlighted by the colored lines.

17 To carry out the simulations of Laplace resonance assembly, we employedthe conventional Bulisch–Stoer algorithm, with an initial time-step ofΔt=0.01 days.

16

The Astrophysical Journal, 894:143 (23pp), 2020 May 10 Batygin & Morbidelli

e.g., Deck & Batygin 2015 for more discussion). Convergentorbital evolution is simulated by applying the type-I migrationtorque (Papaloizou & Larwood 2000) to Europa and Gany-mede. For definitiveness, we adopt a common characteristicmigration timescale for both objects, which maintains theirnonresonant period ratio prior to Europa and Io’s encounterwith the 2:1 commensurability. At the same time, type-Ieccentricity and inclination damping—assumed to operate on atimescale a factor of (h/r)−2=100 times shorter than themigration time (Tanaka & Ward 2004)—is applied to allsatellites.

Figure 11(A) shows the results of our fiducial numericalexperiment, where the migration timescale is set to

= 20,000mig yr. Qualitatively, the satellites follow the sameevolutionary sequence as that outlined in the simulations ofPeale & Lee (2002). Namely, Europa reaches the 2:1commensurability with Io first, leading to the establishmentof a resonant lock. An interplay between resonant dynamics

and continued type-I torque exerted on Europa adiabaticallyexcites the eccentricities of both inner satellites, until thisprocess is stabilized by disk-driven eccentricity damping.Eventually, Ganymede reaches a 2:1 resonance with Europa,and a long-term stable 4:2:1 multiresonant chain is established.As already pointed out in the work of Peale & Lee (2002), theresonance established through convergent migration within thecircumplanetary nebula is a different variant of the Laplaceresonance than the one the satellites occupy today (meaningthat the Laplace angle exhibits asymmetric libration with anappreciable amplitude instead of being tightly confined to180 deg). However, their calculations also demonstrate that assoon as satellite–disk interactions subside and are replaced withconventional tidal evolution, the satellites’ eccentricitiesrapidly decay, leading to the establishment of the observedresonant architecture.The Adiabatic Limit—We note that the migration timescale

adopted in our fiducial numerical experiment exceeds thetheoretical value given by Equation (42) by a factor of ∼5. Wedo not consider this to be a meaningful drawback of our modelbecause it is unlikely that our simplified description of thecircumplanetary disk can predict the actual migration rateexperienced by the Galilean satellites to better than an order ofmagnitude. An arguably more important consideration is thatindependent of any particular formation scenario, the masses ofthe satellites dictate a minimum orbital convergence timescalebelow which the establishment of a long-term stable 4:2:1mean-motion commensurability becomes improbable. Simplyput, this is because adiabatic capture into a mean-motionresonance requires the resonance bandwidth crossing time tosignificantly exceed the libration timescale of the resonantangles. Notably, the former is set by the assumed migrationtimescale while the latter is determined by the satellite masses(see, e.g., Batygin 2015 and the references therein).To quantify the adiabatic limit of the rate of orbital

convergence among Jovian satellites, we repeated the afore-mentioned experiment, reducing the migration timescale by afactor of two, such that = 10,000mig yr. The correspondingresults are depicted in the middle panel (B)) of Figure 11.Although the early stages of this simulation resemble theevolution depicted in Figure 11(A) (in that the satellites do gettemporarily locked into a 4:2:1 resonance), in a matter of a fewthousand years, they break out of this configuration andfollowing a transient period of chaotic dynamics, stabilize in amore compact 8:6:3 resonant chain. Even this configuration,however, is not immutable: approximately 30,000 yr into thesimulation, the system becomes dynamically unstable, andcollisions among satellites ensue shortly thereafter. Thus, weconclude that if the Laplace resonance is indeed primordial, themigration timescale associated with the orbital assembly ofGalilean satellites could not have been much shorter18 than20,000 yr.Overstability—Apart from the migration timescale itself, a

separate constraint on the assembly of the Laplace resonanceconcerns the order in which the observed orbital architecturewas established. Recall that within the context of thesimulations described above, Europa encountered the 2:1mean-motion commensurability with Io before Ganymedejoined the resonant chain. This is due to the fact that within

Figure 11. Formation of the Laplace resonance. Panel (A): migration of Europaand Ganymede toward Io on a = 20,000mig yr timescale. In this simulation,the convergent orbital evolution of the three inner Galilean satellites leads tosequential locking of Io, Europa, and Ganymede into a long-term stable 4:2:1mean-motion commensurability. This sequence of events is consistent with theactual architecture of Jovian satellites. Panel (B): if the convergent migrationtimescale is reduced by a factor of two (such that = 10,000mig yr), the 4:2:1Laplace resonance is rendered to be long-term unstable. In this case, after thesatellites break out of the 4:2:1 commensurately, they temporarily get capturedinto a more compact 8:6:3 resonance. However, this configuration is also long-term unstable, and eventually, a full-fledged orbital instability develops,triggering satellite collisions. Panel (C): a demonstration of resonantoverstability in the Galilean system. If Europa and Ganymede lock into the2:1 commensurability before Io and Europa do, the associated dynamics areover-stable, and in due course, the full system equilibrates within the 6:3:2—rather than the 4:2:1—multiresonant configuration. Cumulatively, thesenumerical experiments point to two independent constraints. First, if theLaplace resonance is primordial, Io and Europa must have locked into the 2:1resonance before Europa and Ganymede approached a 2:1 commensurability.Second, the timescale for convergent migration could not have been muchshorter than ∼20,000 yr.

18 Importantly, independent of the details of the accretion process, thisrequirement for a relatively long migration timescale necessitates a low-masscircumplanetary disk.

17

The Astrophysical Journal, 894:143 (23pp), 2020 May 10 Batygin & Morbidelli

the framework of our model, satellite formation is envisioned tooccur successively rather than simultaneously. Indeed, had allthree satellites emerged within the circumplanetary disk at thesame time, the Europa–Ganymede resonance would have beenestablished first, since Ganymede is more massive and wouldhave experienced more rapid orbital decay. As it turns out,sequential formation of satellites is not simply a naturaloutcome of our theoretical picture (as depicted in Figure 9)—itis a veritable requirement of the observed resonant dynamics.

An intriguing aspect of disk-driven resonant encounters isthat the long-term stability of the ensuing resonance can becompromised by the same dissipation that leads to itsestablishment. This effect—known as resonant overstability—exhibits a strong dependence on the satellite mass ratio andmanifests in systems where the outer secondary body is moremassive than the inner (Goldreich & Schlichting 2014; Deck &Batygin 2015; Xu et al. 2018). To this end, the analyticcriterion for overstability (see Figure 3 of Deck & Batygin2015) suggests that the factor of ∼3 difference betweenthe masses of Ganymede and Europa is sufficient to renderthe 2:1 resonance unstable, if the satellite pair encounters it inisolation. In other words, overstability of resonant dynamicsindicates that the observed 4:2:1 Laplace resonance could nothave been established if Ganymede and Europa locked intoresonance before Europa and Io did.

To confirm this anticipation, we repeated the abovenumerical experiment, restoring mig to 20,000 yr, but onlyapplying the migration torque to Ganymede (conversely,eccentricity damping torque was applied to all satellites asbefore). This choice ensured that Ganymede would encounterthe 2:1 resonance with Europa first, since in this experiment,Europa experiences no explicit disk-driven migration. Theresults of this simulation—depicted in Figure 11(C)—followedanalytic expectations precisely: upon entering the 2:1 reso-nance, overstable librations ensued, propelling Europa andGanymede to break out of the 2:1 resonance before theestablishment of the 4:2:1 resonant chain. Eventually, con-vergent migration did drive the system into a multiresonantconfiguration, but it was characterized by a 6:3:2 period ratio.Indeed, the Io–Europa–Ganymede Laplace resonance appearsto have been built from the inside out.

6.4. A Final Wave of Accretion

Up until this point, we were primarily concerned with theconglomeration and migration of the inner three Galileansatellites within the gaseous circumplanetary disk. But whathappens when the photoevaporation front reaches the giantplanets’ orbits and the gas is removed? One trivial consequenceof gas removal is that the metallicity of the system ¥everywhere in the disk. Referring back to Equation (31), thiswould imply that the Q•1 condition would be satisfied at allorbital radii (i.e., not just the outer disk as shown in Figure 5),implying the onset of a final wave of satellitesimal formation.Accordingly, let us now consider the growth of a satelliteembryo within this gas-free environment, with an eye towardquantifying the formation of Callisto and Titan.

In terms of basic characteristics, Callisto and Titan sharemany similarities. Both have orbital periods slightly in excessof two weeks (corresponding to approximately 3.5% and 2% ofJupiter and Saturn’s respective Hill spheres). The physical radiiand masses of the satellites are also nearly identical (although,when normalized by the masses of their host planets, Titan is

larger than Callisto by a factor of ∼4). Finally, measurementsof the satellites’ axial moments of inertia through spacecraftgravity data point to the distinct possibility that these satellitesmay be only partially differentiated (Anderson et al. 2001; Iesset al. 2010; see however Gao & Stevenson 2013), implying aformation timescale that exceeds ∼0.5 Myr (Barr &Canup 2008). Accordingly, let us now examine if the post-nebular phase of our model can naturally generate bodiessharing some of Callisto and Titan’s attributes. For definitive-ness, in the remainder of this section, we will concentrate onthe formation of Callisto, although the applicability of thecalculations to Titan is also implied.We begin by envisioning proto-Callisto as a satellite embryo

embedded in a disk of icy debris. When submerged in a sea ofsolid material, the embryo interacts with its environment bygravitationally stirring the satellitesimal swarm (Safronov1969). This process has two direct consequences: collisionswith small bodies lead to steady accretion, while asymmetricscattering of debris facilitates transfer of angular momentum(Ida et al. 2000; Kirsh et al. 2009). Therefore, in a gas-freeenvironment, growth of the embryo must occur concurrentlywith satellitesimal-driven migration. Although some analyticunderstanding of the associated physics exists (Minton &Levison 2014), N-body simulations provide the clearestillustration of the ensuing dynamical evolution. Correspond-ingly, in an effort to maintain a closer link with othercalculations presented in this section, we continue on with anumerical approach.Our simulation setup essentially constituted a stripped-down,

purely gravitational variant of the calculations presented inSection 6.2. More specifically, a disk of 1000 supersatellite-simals comprising = ´ -M M2 10disk

4◦ was initialized

between =r R0.03in H and =r R0.3out H, together with a singlesatellite embryo residing in the outer disk at =r R0.25 H. Thisswarm of satellitesimals is envisioned to have coalesced fromthe remainder of debris left behind in the aftermath of theaccretion of Io, Europa, and Ganymede, as well as newsatellitesimals, formed from the leftover dust in a wave ofgravitational collapse triggered by the dissipation of the gas.Owing to the envisioned lack of residual hydrogen and heliumwithin the system, both the effects of aerodynamic drag as wellas type-I orbital migration were assumed to be negligible.Moreover, with no gas to dynamically cool the system, weassumed that the initial velocity dispersion of the satellitesimalswas set by gravitational self-stirring and was thereforecomparable to the escape velocity of the small bodiesá ñ ~v vesc. All other details of the numerical calculations wereidentical to those reported in Section 6.2.As before, 12 numerical experiments were carried out but

with the integration time increased to 10Myr. We note that theassumed inner edge of the planetesimal disk lies slightlyinterior to Callisto’s current orbital semimajor axis andapproximately coincides with the location of Ganymede’sexterior 2:1 resonance. In light of this correspondence, it isnatural to expect that as proto-Callisto approaches its finalorbit, satellitesimals at the inner edge of the particle disk willexperience a complex interplay of perturbations arising fromCallisto itself as well as from Ganymede. Nevertheless, tosubdue the already formidable computational costs andmaintain a reasonably long time-step, we disregard theexistence of the inner three Galilean satellites in thesesimulations. Although this assumption does not pose a

18

The Astrophysical Journal, 894:143 (23pp), 2020 May 10 Batygin & Morbidelli

significant problem throughout most of the simulation domain,the dynamical evolution exhibited by the system close to rinshould indeed be viewed as being highly approximate.

A variant of Figure 9 pertinent to the formation of Callisto isshown in Figure 12. Initial conditions characterized by a Safronovnumber of order unity are depicted in panel (A)). Althoughembryo growth accompanied by gravitational stirring of the diskstarts immediately (panel (B)), it proceeds very slowly at first. Acomparatively rapid phase of accretion and satellitesimal-drivenmigration ensues 100,000 yr into the simulation (panel (C)) butterminates at 200,000 yr, with Callisto having achievedapproximately half of its mass (panel (D)). Owing to an excitedorbital distribution, subsequent growth unfolds on an exception-ally long timescale (panel (E)), such that Callisto only achieves itsfull mass 8Myr into the simulation (panel (F)).

We note that if left unperturbed, the remaining satellitesimalswarm beyond r R0.1 H would eventually coalesce intoadditional satellites. However, Deienno et al. (2014) andNesvorný et al. (2014) have demonstrated that any objectsbeyond the orbits of Callisto and Iapetus are readilydestabilized by planetary flybys that transpire during the solarsystem’s transient phase of dynamical instability. All satellitesinterior to Titan, on the other hand, can, in principle, beaccounted for by the ring-spreading model of Crida & Charnoz(2012; see also the work of Charnoz et al. 2011). Consequently,the radial extent of the regular satellite systems of Jupiter and

Saturn likely reflect a combination of processes includinggravitationally focused pair-wise accretion, orbital migration,and external dynamical sculpting.

7. Discussion

Although subject to nearly continuous astronomical monitor-ing for centuries, the natural satellites of Jupiter and Saturn haveonly come into sharper focus within the last 40 yr. Theunprecedented level of detail unveiled by the Voyager flybys(Smith et al. 1979a, 1979b) as well as the Galileo/Cassiniorbiters (Greeley et al. 1998; Moore et al. 1998; Pappalardo et al.1998; Elachi et al. 2005; Stofan et al. 2007; Hayes 2016) hasincited a veritable revolution in our understanding of thesefaraway moons, once and for all transforming them from celestialcuriosities into bona fide extraterrestrial worlds. This ongoingparadigm shift—sparked during the latter half of the last century—is poised to persist in the coming decades, as Europa Clipper,JUICE, and Dragonfly missions,19 along with ground-basedphotometric/spectroscopic observations (Trumbo et al.2017, 2019; de Kleer et al. 2019a, 2019b), continue to deepenour insight into their geophysical structure. Importantly, all ofthese developments have added a heightened element of

Figure 12. Formation of Callisto in a gas-free satellitesimal swarm. A similar accretion scenario can be envisioned for Titan. A disk of debris, comprising= ´ -M M2 10disk

4◦ was initialized between =r R0.03in H and =r R0.3out H, with a Safronov number of order unity: Θ∼1 (panel (A)). A single satellite seed—

depicted in red—is introduced at =r R0.25 H. Unlike the results reported in Figure 9, in absence of dissipative effects associated with the presence of thecircumplanetary disk, the initial phase of satellite conglomeration proceeds slowly. Correspondingly, 50,000 yr into the simulation, the embryo has only acquired 5%of Callisto’s mass (panel (B)). In the following 100,000 yr, growth temporarily accelerates, and in concert with the accretion process, the satellite embryo migratesinward by scattering satellitesimals (panel (C)). By the 200,000 year mark, the satellite embryo is a factor of two less massive than Callisto, but due to the orbitalexcitation of neighboring debris, satellitesimal-driven migration effectively grinds to a halt (panel (D)). Subsequently, the rate of accretion slows down dramatically,such that 2 million years after the start of the simulation, proto-Callisto has only reached about three-quarters of its terminal mass (panel (E)). The embryo finallyachieves Callisto’s actual mass after 8 million years of evolution (panel (F)).

19 JUICE, Europa Clipper, and Dragonfly missions are expected to launch in2022, 2025, and 2026, respectively.

19

The Astrophysical Journal, 894:143 (23pp), 2020 May 10 Batygin & Morbidelli

intrigue to the unfaltering quest to unravel the origins of naturalsatellites within the solar system (Peale 1999).

In parallel with in-situ exploration of the Sun’s planetaryalbum, detailed characterization of gas flow within youngextrasolar nebulae (Isella et al. 2019; Teague et al. 2019) hasbegan to illuminate the intricate physical processes that operateconcurrently with the final stages of giant planet accretion.Coupled with high-resolution hydrodynamical simulations offluid circulation within planetary Hill spheres (Tanigawa et al.2012; Morbidelli et al. 2014; Szulágyi et al. 2014, 2016;Lambrechts et al. 2019), these results have painted an updatedportrait of the formation and evolution of circumplanetarydisks. While this newly outlined picture has been instrumentalto the successful interpretation of modern observations, it hasalso brought to light a series of puzzles that remain elusivewithin the context of the standard model of satellite formation(Canup & Ward 2002, 2006). In particular, the physical processthat underlies the agglomeration of sufficiently large quantitiesof dust within the circumplanetary disks, the mechanism forconversion of this dust into satellite building blocks, and theprimary mode by which satellitesimals accrete into full-fledgedsatellites have remained imperfectly understood.

In this paper, we have presented our attempt at answeringthese questions from first principles. Let us briefly summarizeour proposed scenario.

7.1. Key Results

Inspired by the aforementioned observational and computa-tional results, we have considered the conglomeration ofsatellites within a vertically fed, steady-state H/He decretiondisk that encircles a young giant planet. Owing to pronouncedpressure support, gas circulation within this disk is notably sub-Keplerian and is accompanied by a (viscously driven) radialoutflow in the midplane. Although the system originatesstrongly depleted in heavy elements, its metallicity is envisagedto increase steadily in time. More specifically, our calculationsshow that s•10 mm dust grains are readily trapped withinour model circumplanetary disk, thanks to a hydrodynamicequilibrium that ensues from a balance between energy gainsand losses associated with the radial updraft and azimuthalheadwind, respectively. While it may be impossible todefinitively prove that this process truly operated in gaseousnebulae that encircled Jupiter and Saturn during the solarsystem’s infancy, our theoretical picture exhibits a remarkabledegree of consistency with the recent observations of Bae et al.(2019), which demonstrate that the circumplanetary disk inorbit of the young giant planet PDS 70 c is enriched in dust bymore than an order of magnitude compared with the expectedbaseline metallicity.

As the dust-to-gas ratio of the system grows, the solidsubdisk progressively settles toward the midplane, eventuallybecoming thin enough for gravitational fragmentation to ensue(Goldreich & Ward 1973). Correspondingly, large-scalegravitational collapse of the dust layer generates a satellitesimaldisk containing ~ ´ -M M6 10disk

4◦ worth of material

between ~ R0.1 H and ~ R0.3 H. The velocity dispersion ofthe resulting satellitesimal swarm is heavily damped byaerodynamic drag originating from the gas, allowing forefficient capture of satellitesimals by an emerging satelliteembryo (Safronov 1969; Adachi et al. 1976). Assisted bygravitational focusing, this embryo continues accreting until itbecomes massive enough to raise significant wakes within its

parent nebula. The gravitational back-reaction of the spiraldensity waves upon the satellite results in orbital decay (e.g.,Tanaka et al. 2002), terminating further growth of the newlyformed satellite by removing it from the satellitesimal feedingzone. Eventually, the satellite reaches the inner boundary of thedisk and halts its migratory trek.An important attribute of the above picture is that as the satellite

exists the feeding zone by orbital migration, it leaves behind adynamically excited orbital distribution of satellitesimals. Withnothing to facilitate continued gravitational stirring, however,satellitesimal orbits recircularize and collapse back down to theequatorial plane under the action of aerodynamic drag. Thesatellite formation process then restarts, generating a secondembryo. Eventually, this embryo also grows to its terminal radius,dictated by a near-equality of the migration and mass-doublingtimescales (see also Canup & Ward 2006), and subsequently alsoexits the satellitesimal swarm. If the proceeding satellite is retainedwithin the inner region of the disk, convergent migration of thebodies facilitates locking into a mean-motion resonance. If thissequence of events occurs more than twice, a resonant chain—akin to that exhibited by the three inner Galilean moons—can benaturally generated (Peale & Lee 2002). Our calculations furtherdemonstrate that due to constraints associated with resonantoverstability (Goldreich & Schlichting 2014; Deck & Batygin2015), the Laplace resonance must have been assembled from theinside out and that the timescale for orbital convergence musthave exceeded 20,000mig yr.In principle, we can imagine that the process of sequential

satellite generation continues until the gas is abruptly removedby photoevaporation of the circumstellar nebula (Owen et al.2012). To this end, we note that the gravitational bindingenergy of a hydrogen molecule in orbit around Jupiter at~r R0.01 H is approximately equal to that of a hydrogen

molecule in orbit around the Sun at r∼5 au. This means thatthe same solar photons that successfully eject gas from theSun’s potential well at Jupiter’s orbit can also expel gas fromthe Jovian potential well at a distance comparable to thetruncation radius. In turn, this implies that the removal of thegaseous components of the circumplanetary nebula and thecircumstellar disk must occur simultaneously.Driven by a sharp increase in effective disk metallicity, the

remaining dust subdisk fragments into satellitesimals, settingthe stage for the accretion of the final satellite embryo.However, unlike the comparatively rapid mode of satelliteformation described above, in this gas-free environment,embryo growth proceeds on a multimillion-year timescale,leading to only partial differentiation of the resulting body(Barr & Canup 2008). The mechanism of inward migration isalso distinct in that it is facilitated by asymmetric scattering ofdebris rather than tidal interactions with the gas (Kirsh et al.2009). For the specific purposes of this study, we consider theslow conglomeration process to be relevant to the formation ofCallisto (and perhaps Titan). As a concluding step to thenarrative, we invoke the effects of planet–planet scatteringduring the transient phase of giant planet instability to dispersethe remaining satellitesimal disk, leaving only the deepestsegments of the planetary Hill spheres to host large naturalsatellites (Deienno et al. 2014; Nesvorný et al. 2014).

7.2. Jupiter versus Saturn

Despite sharing some basic properties, the satellite systemsof Jupiter and Saturn are far from identical, and it is worthwhile

20

The Astrophysical Journal, 894:143 (23pp), 2020 May 10 Batygin & Morbidelli

to contemplate how the differences between them came to be.In Section 6, we asserted that the formation narratives ofCallisto and Titan may be similar, leaving open the question ofwhy Saturn does not possess an equivalent system of large,multiresonant (Galilean-type) inner moons. Within the frame-work of our model, two separate explanations for this disparitycan be conjured up. Perhaps most simply, we can envision ascenario where Saturn’s circumplanetary nebula never achievedthe requisite metallicity for large-scale fragmentation of thesolid subdisk (after all, agglomeration an overall metallicity inexcess of 0.1 is not guaranteed). This would have delayedthe process of satellitesimal formation until the dissipation ofthe gas, bringing the initial conditions of the Saturnian systemin line with those assumed in Section 6.

Alternatively, we may attribute the difference betweenJovian and Kronian systems to a disparity in the host planets’ancient dynamos. That is, if Saturn’s magnetosphere wasinsufficiently prominent to truncate the circumplanetary diskoutside of the Roche radius, any inward-migrating satelliteswould have been tidally disrupted, leaving Titan as “the last ofthe Mohicans” (Canup & Ward 2006). We will resist the urgeto speculate as to which of these imagined solutions may bemore likely, and simply limit ourselves to pointing out thatwhile the latter scenario would imply a primordial origin forSaturn’s rings (Canup 2010; Crida et al. 2019), the formerpicture is more consistent with the recently proposed “youngrings” hypothesis20 (Asphaug & Reufer 2013; Ćuk et al. 2016;Dubinski 2019).

7.3. Criticisms and Future Directions

Although the calculations summarized above outline asequential narrative for the formation of giant planet satellites,much additional work remains to be done before our model canbe considered complete on a detailed level. Accordingly, let usnow propose a series of criticisms of the envisioned scenarioand delineate some avenues for future development of thetheoretical picture.

Arguably the most basic critique of our model concerns acoincidence of timescales. More specifically, we have imaginedthat satellite formation—despite requiring hundreds of thousandsof years to complete—unfolds in a steady-state circumplanetarydisk that encircles an already-assembled giant planet. In order forthis picture to hold, two criteria have to be satisfied. First, theappearance of the circumplanetary disk must concur with theconcluding epochs of Jupiter and Saturn’s respective phases ofrapid gas agglomeration (although some theoretical evidence thatsupports this notion already exists within the literature,additional work is undoubtedly required; Szulágyi et al. 2016;Lambrechts et al. 2019). Second, after concluding theirformation sequences, the solar system’s giant planets wouldhave had to reside inside the protosolar nebula for an extendedperiod of time without experiencing appreciable additionalgrowth.21 This would imply the existence of a yet-to-be-characterized process that suspends runaway accretion of agiant planet within a long-lived circumstellar nebula. Althougha physical mechanism that could robustly regulate the terminalmasses of giant planets remains elusive (Morbidelli et al. 2014;Ginzburg & Chiang 2019), the emergence of an extended

planetary magnetosphere due to an enhanced surface luminos-ity presents one possible option that merits further considera-tion (Batygin 2018; Cridland 2018).A distinct issue relates to the dust–gas equilibrium derived in

Section 3. In particular, the grain radii encapsulated by theheadwind-updraft balance outlined in this work are relativelylimited (i.e., s•∼0.1–10mm), and this restriction illuminatesgrain growth as a potential pathway for dust particles to break outof equilibrium. While the cycle of nucleation followed bysublimation of volatile species inside the circumplanetary disk’sice-line discussed in Section 3 should modulate the grain radii toremain within the aforementioned range, the viability of thismechanism remains to be demonstrated quantitatively. Finally, theprocess of satellitesimal formation within the circumplanetary diskdeserves more careful scrutiny. That is, although the energy-limited settling of dust followed by gravitational fragmentationinvoked in Section 4 provides a particularly simple scenario forconversion of dust into satellite building blocks, it is entirelyplausible that a detailed examination of high-metallicity dust–gasdynamics will reveal a more exotic mode of satellitesimalformation that has eluded our analysis.

7.4. Planet versus Satellite Formation

We began this paper by highlighting a similarity betweendetected systems of extrasolar super-Earths and the moons ofgiant planets. Correspondingly, let us conclude this work with abrief comment on the relationship between the machinery thatunderlies our proposed theory of satellite accretion and thestandard theory of planet formation (Armitage 2010). Undeni-ably, some analogies must exist between our model and thestandard narrative of super-Earth conglomeration, since inwardorbital migration, which is terminated by a steep reduction inthe gas surface density due to magnetic truncation of the disk(e.g., Masset et al. 2006; Izidoro et al. 2019), plays a notablerole in both cases. The parallels, however, may end there.Recall that within the context of our picture, satellite building

blocks are envisioned to form via the Goldreich–Ward mechanism,which is in turn facilitated by a hydrodynamic equilibrium that canonly be achieved in a decretion disk. This view is in stark contrastwith the now widely accepted model of planetesimal formation,which invokes the streaming instability (although both processesculminate in gravitational collapse and generate ~ 100 kmobjects; Youdin & Goodman 2005; Johansen & Youdin 2007).Moreover, pebble accretion, which is increasingly believed to drivethe conglomeration of giant planet cores and super-Earths alike(Lambrechts & Johansen 2012; Bitsch et al. 2015), is unlikely toplay the leading role in facilitating satellitesimal growth. Instead,satellite accretion proceeds via pair-wise collisions amongsatellitesimals, entailing a closer link to the physics of terrestrialplanet formation (Lissauer 1993; Hansen 2009; Walsh &Levison 2016; Ogihara et al. 2018) than anything else.Cumulatively, these contrarieties suggest that the model outlinedherein cannot be readily applied to circumstellar disks, and thearchitectural similarities between solar system satellites and short-period extrasolar planets are likely to be illusory.

We are thankful to Katherine de Kleer, Darryl Seligman, PhilHopkins, Mike Brown, and Christopher Spalding for insightfuldiscussions. We thank Thomas Ronnet for providing a carefuland insightful review of the manuscript. K.B. is grateful to theDavid and Lucile Packard Foundation and the Alfred P. SloanFoundation for their generous support.

20 While plausible, a qualitative mechanism that could feasibly generate therings within the last few tens of millions of years remains elusive.21 Notably, such a sequence of events is required for the so-called Grand Tackscenario of early solar system evolution (Walsh et al. 2011).

21

The Astrophysical Journal, 894:143 (23pp), 2020 May 10 Batygin & Morbidelli

Appendix ADust Equilibrium from NSH Drift

For simplicity, the analysis presented in Section 3 wascarried out under the assumption that the cumulative back-reaction of solid dust upon gas is negligible. A more generaldescription of dust–gas interactions within a nearly Kepleriandisk is provided by the Nakagawa–Sekiya–Hayashi drift. In alocally Cartesian Keplerian ( f= =x r y r,ˆ ˆ ˆ ˆ ) frame, thestandard equations pertinent to the dust component of thesystem take the form (Nakagawa et al. 1986)

h tt

ht

=-+ +

=-+

+ ++f

vv

x

vv

v y

2

11

1. A1

r •K

2 2

•K

2 2 K

( )ˆ

( )( )

ˆ ( )

Qualitatively, the above expressions simply state that theprimary outcome of gas–dust coupling is the inward drift ofsolids that is compensated by the outward expulsion of gas.

Importantly, these equations were derived assuming that theunperturbed azimuthal velocity of the gas is h= -fv v1 K( )and that the unperturbed radial velocity is null. Conversely, ifbaseline radial velocity of the gas is vr, by analogy with thesecond equation for the dust above, we have

h tt t

= -+ +

++

+ +

v

v vx

2

1

1

1. A2r

r•

K2 2 2 2( )

( )( )

ˆ ( )

Setting the rhs of this equation equal to zero, we obtain theequilibrium Stokes number that yields =v 0r • :

th

= + v

v21 . A3req

K( ) ( )( )

Appendix B3D Bondi Accretion and the Transition to the Hill Regime

For convenience, in Section 5.1, we expressed the Bondi–Hill crossover mass,t (Equation (38)), as well as the rate ofpebble accretion itself (Equation (39)) in terms of global diskproperties. Here, we outline the derivation of these expressions.

The Bondi and Hill capture radii RcB and Rc

H are given byEquations (36) and (37), respectively. As a starting step, weraise both quantities to the sixth power and set them equal toone another. Accordingly, =R Rc

B 6cH 6( ) ( ) gives

h h+

=+ r v

M v

r v

v

25 1

9

1

8. B1r r

2 6 2mid

2

2 2K2

3 3 3 3mid

6

2K9

( ) ( ) ( )◦

Rearranging for, we obtain

h=

+

r v

M v

200

9 1. B2

r

4 3K7

3 2mid

4( )( )

◦

Substituting p Sr M2 for v1 r and multiplying the numeratoras well as the denominator by M r◦ , yields

p h=

S+

Mr

M

r

Mv

v

r

400

9 1. B3

3

3 3

2

K6 K

4

mid4( )

( )◦◦

Finally, canceling r M 3( ( ))◦ and vK6 while consolidating

v rK into Ω, we arrive at Equation (38),

p h=

+S W

r

MM

400

9 1. B4t

mid

4 2( )◦

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

The rate of accretion in the Bondi regime is derived in asimilar manner. We begin by noting that the product of Rc

B 2( )and Δv is

hD =

+ R v

r v

v

1

2. B5r

cB 2 mid

K2

( ) ( ) ( )

Meanwhile, S = S• and a= h r h r2• ( ) ( ). There-fore,

aS

=S -

h r

h

r

2. B6•

•

3 1

( )⎜ ⎟⎛⎝

⎞⎠

Multiplying the two expressions together while introducing afactor of M◦ in both the numerator and the denominator, wehave

h a

DS

=

´S + -

R vh

M

M M

M

r

r v

v

h

r

1

2

2. B7r

cB 2 •

•

mid

K2

1 3

( )

( ) ( )

◦

◦ ◦

◦

⎜ ⎟⎛⎝

⎞⎠

Canceling M r◦ with v1 K2 and multiplying the above

expression by a factor of p 2 , we obtain Equation (39),

p

hpa

=S

D

=+

S-

d

d t hR v

M

h

rr v

2

1

2

1. B8r

B 3D

•

•cB 2

mid1 3

( )

( )◦

⎜ ⎟

⎜ ⎟

⎛⎝

⎞⎠

⎛⎝

⎞⎠

ORCID iDs

Konstantin Batygin https://orcid.org/0000-0002-7094-7908

References

Adachi, I., Hayashi, C., & Nakazawa, K. 1976, PThPh, 56, 1756Anderson, J. D., Jacobson, R. A., McElrath, T. P., et al. 2001, Icar, 153, 157Armitage, P. J. 2010, Astrophysics of Planet Formation (Cambridge:

Cambridge Univ. Press)Asphaug, E., & Reufer, A. 2013, Icar, 223, 544Bae, J., Zhu, Z., Baruteau, C., et al. 2019, ApJL, 884, L41Barr, A. C., & Canup, R. M. 2008, Icar, 198, 163Batygin, K. 2015, MNRAS, 451, 2589Batygin, K. 2018, AJ, 155, 178Benz, W., & Asphaug, E. 1999, Icar, 142, 5Bitsch, B. 2019, A&A, 630, A51Bitsch, B., Lambrechts, M., & Johansen, A. 2015, A&A, 582, A112Boss, A. P. 1997, Sci, 276, 1836Canup, R. M. 2010, Natur, 468, 943Canup, R. M., & Ward, W. R. 2002, AJ, 124, 3404Canup, R. M., & Ward, W. R. 2006, Natur, 441, 834Canup, R. M., & Ward, W. R. 2009, Europa (Tucson: Univ. Arizona Press), 59Chambers, J. E. 1999, MNRAS, 304, 793Charnoz, S., Crida, A., Castillo-Rogez, J. C., et al. 2011, Icar, 216, 535Crida, A., & Charnoz, S. 2012, Sci, 338, 1196Crida, A., Charnoz, S., Hsu, H.-W., et al. 2019, NatAs, 3, 967Crida, A., Morbidelli, A., & Masset, F. 2006, Icar, 181, 587Cridland, A. J. 2018, A&A, 619, A165Ćuk, M., Dones, L., & Nesvorný, D. 2016, ApJ, 820, 97

22

The Astrophysical Journal, 894:143 (23pp), 2020 May 10 Batygin & Morbidelli

Cuzzi, J. N., Dobrovolskis, A. R., & Champney, J. M. 1993, Icar, 106, 102de Kleer, K., de Pater, I., Molter, E. M., et al. 2019a, AJ, 158, 29de Kleer, K., Nimmo, F., & Kite, E. 2019b, GeoRL, 46, 6327Deck, K. M., & Batygin, K. 2015, ApJ, 810, 119Deienno, R., Nesvorný, D., Vokrouhlický, D., et al. 2014, AJ, 148, 25Drążkowska, J., & Szulágyi, J. 2018, ApJ, 866, 142Dubinski, J. 2019, Icar, 321, 291Dubrulle, B., Morfill, G., & Sterzik, M. 1995, Icar, 114, 237Dullemond, C. P., Birnstiel, T., Huang, J., et al. 2018, ApJL, 869, L46Elachi, C., Wall, S., Allison, M., et al. 2005, Sci, 308, 970Estrada, P. R., Mosqueira, I., Lissauer, J. J., et al. 2009, Europa (Tucson: Univ.

Arizona Press), 27Fung, J., & Chiang, E. 2016, ApJ, 832, 105Gao, P., & Stevenson, D. J. 2013, Icar, 226, 1185Ginzburg, S., & Chiang, E. 2019, MNRAS, 487, 681Ginzburg, S., & Chiang, E. 2020, MNRAS, 491, L34Goldreich, P. 1965, MNRAS, 130, 159Goldreich, P., Lithwick, Y., & Sari, R. 2004, ApJ, 614, 497Goldreich, P., & Schlichting, H. E. 2014, AJ, 147, 32Goldreich, P., & Tremaine, S. 1980, ApJ, 241, 425Goldreich, P., & Ward, W. R. 1973, ApJ, 183, 1051Greeley, R., Sullivan, R., Klemaszewski, J., et al. 1998, Icar, 135, 4Hansen, B. M. S. 2009, ApJ, 703, 1131Haugbølle, T., Weber, P., Wielandt, D. P., et al. 2019, AJ, 158, 55Hayashi, C. 1981, PThPS, 70, 35Hayes, A. G. 2016, AREPS, 44, 57Henrard, J. 1982, CeMec, 27, 3Ida, S., Bryden, G., Lin, D. N. C., et al. 2000, ApJ, 534, 428Ida, S., Guillot, T., & Morbidelli, A. 2016, A&A, 591, A72Ida, S., Ueta, S., Sasaki, T., et al. 2020, arXiv:2003.13582Iess, L., Rappaport, N. J., Jacobson, R. A., et al. 2010, Sci, 327, 1367Isella, A., Benisty, M., Teague, R., et al. 2019, ApJL, 879, L25Isella, A., & Turner, N. J. 2018, ApJ, 860, 27Izidoro, A., Bitsch, B., Raymond, S. N., et al. 2019, arXiv:1902.08772Johansen, A., & Lambrechts, M. 2017, AREPS, 45, 359Johansen, A., Oishi, J. S., Mac Low, M.-M., et al. 2007, Natur, 448, 1022Johansen, A., & Youdin, A. 2007, ApJ, 662, 627Johansen, A., Youdin, A., & Mac Low, M.-M. 2009, ApJL, 704, L75Kane, S. R., Hinkel, N. R., & Raymond, S. N. 2013, AJ, 146, 122Kant, I. 1755, Allgemeine Naturgeschichte und Theorie des Himmels

(Königsberg: Petersen)Kirsh, D. R., Duncan, M., Brasser, R., et al. 2009, Icar, 199, 197Kokubo, E., & Ida, S. 1998, Icar, 131, 171Kuwahara, A., & Kurokawa, H. 2020, A&A, 633, A81Lambrechts, M., & Johansen, A. 2012, A&A, 544, A32Lambrechts, M., Johansen, A., & Morbidelli, A. 2014, A&A, 572, A35Lambrechts, M., Lega, E., Nelson, R. P., et al. 2019, A&A, 630, A82Laplace, P. 1796, Exposition du Ssystème du Monde (Paris: Cerie-Social)Lari, G., Saillenfest, M., & Fenucci, M. 2020, arXiv:2001.01106Laughlin, G., & Lissauer, J. J. 2015, arXiv:1501.05685Lee, U., Osaki, Y., & Saio, H. 1991, MNRAS, 250, 432Levison, H. F., & Duncan, M. J. 2000, AJ, 120, 2117Lin, M.-K. 2019, MNRAS, 485, 5221Lissauer, J. J. 1993, ARA&A, 31, 129Liu, B., Lambrechts, M., Johansen, A., et al. 2019, A&A, 632, A7Lunine, J. I., & Stevenson, D. J. 1982, Icar, 52, 14Lynden-Bell, D., & Pringle, J. E. 1974, MNRAS, 168, 603Lyra, W., Johansen, A., Zsom, A., et al. 2009, A&A, 497, 869Masset, F. S., Morbidelli, A., Crida, A., et al. 2006, ApJ, 642, 478Mestel, L. 1963, MNRAS, 126, 553Miguel, Y., & Ida, S. 2016, Icar, 266, 1Millholland, S., Wang, S., & Laughlin, G. 2017, ApJL, 849, L33Minton, D. A., & Levison, H. F. 2014, Icar, 232, 118Mohanty, S., & Shu, F. H. 2008, ApJ, 687, 1323Moore, J. M., Asphaug, E., Sullivan, R. J., et al. 1998, Icar, 135, 127Morbidelli, A. 2020, A&A, in press (arXiv:2004.04942)

Morbidelli, A., Bottke, W. F., Nesvorný, D., et al. 2009, Icar, 204, 558Morbidelli, A., & Nesvorny, D. 2012, A&A, 546, A18Morbidelli, A., & Raymond, S. N. 2016, JGRE, 121, 1962Morbidelli, A., Szulágyi, J., Crida, A., et al. 2014, Icar, 232, 266Mosqueira, I., Estrada, P., & Turrini, D. 2010, SSRv, 153, 431Mosqueira, I., & Estrada, P. R. 2003a, Icar, 163, 198Mosqueira, I., & Estrada, P. R. 2003b, Icar, 163, 232Nakagawa, Y., Sekiya, M., & Hayashi, C. 1986, Icar, 67, 375Nesvorný, D., Vokrouhlický, D., Deienno, R., et al. 2014, AJ, 148, 52Ogihara, M., Kokubo, E., Suzuki, T. K., et al. 2018, A&A, 612, L5Øksendal, B. 2013, Stochastic Differential Equations (Berlin: Springer)Ormel, C. W., & Klahr, H. H. 2010, A&A, 520, A43Owen, J. E., Clarke, C. J., & Ercolano, B. 2012, MNRAS, 422, 1880Paardekooper, S.-J., & Johansen, A. 2018, SSRv, 214, 38Papaloizou, J. C. B., & Larwood, J. D. 2000, MNRAS, 315, 823Pappalardo, R. T., Head, J. W., Collins, G. C., et al. 1998, Icar, 135, 276Peale, S. J. 1976, ARA&A, 14, 215Peale, S. J. 1999, ARA&A, 37, 533Peale, S. J., & Lee, M. H. 2002, Sci, 298, 593Pinilla, P., Benisty, M., & Birnstiel, T. 2012, A&A, 545, A81Poon, S. T. S., Nelson, R. P., Jacobson, S. A., et al. 2020, MNRAS, 491,

5595Press, W. H., Teukolsky, S. A., Vetterling, W. T., et al. 1992, Numerical

Recipes in FORTRAN. The Art of Scientific Computing (Cambridge:Cambridge Univ. Press)

Pringle, J. E. 1991, MNRAS, 248, 754Ronnet, T., & Johansen, A. 2020, A&A, 633, A93Rosenthal, M. M., & Murray-Clay, R. A. 2019, arXiv:1908.06991Safronov, V. S. 1960, AnAp, 23, 979Safronov, V. S. 1969, Evolution of the Protoplanetary Cloud and Formation of

the Earth and Planets (Moscow: Nauka Press), Trans. NASA TTF 677, 1972Sasaki, T., Stewart, G. R., & Ida, S. 2010, ApJ, 714, 1052Sekiya, M. 1998, Icar, 133, 298Seligman, D., Hopkins, P. F., & Squire, J. 2019, MNRAS, 485, 3991Shakura, N. I., & Sunyaev, R. A. 1973, A&A, 500, 33Smith, B. A., Soderblom, L. A., Beebe, R., et al. 1979a, Sci, 206, 927Smith, B. A., Soderblom, L. A., Johnson, T. V., et al. 1979b, Sci, 204, 951Squire, J., & Hopkins, P. F. 2018, ApJL, 856, L15Stofan, E. R., Elachi, C., Lunine, J. I., et al. 2007, Natur, 445, 61Szulágyi, J., Masset, F., Lega, E., et al. 2016, MNRAS, 460, 2853Szulágyi, J., Morbidelli, A., Crida, A., et al. 2014, ApJ, 782, 65Takeuchi, T., & Lin, D. N. C. 2002, ApJ, 581, 1344Tanaka, H., Takeuchi, T., & Ward, W. R. 2002, ApJ, 565, 1257Tanaka, H., & Ward, W. R. 2004, ApJ, 602, 388Tanigawa, T., Ohtsuki, K., & Machida, M. N. 2012, ApJ, 747, 47Teague, R., Bae, J., & Bergin, E. A. 2019, Natur, 574, 378Thompson, S. E., Coughlin, J. L., Hoffman, K., et al. 2018, ApJS, 235, 38Toomre, A. 1964, ApJ, 139, 1217Trumbo, S. K., Brown, M. E., Fischer, P. D., et al. 2017, AJ, 153, 250Trumbo, S. K., Brown, M. E., & Hand, K. P. 2019, SciA, 5, aaw7123Varnière, P., & Tagger, M. 2006, A&A, 446, L13Walsh, K. J., & Levison, H. F. 2016, AJ, 152, 68Walsh, K. J., Morbidelli, A., Raymond, S. N., et al. 2011, Natur, 475, 206Ward, W. R. 1997, Icar, 126, 261Weber, P., Benítez-Llambay, P., Gressel, O., et al. 2018, ApJ, 854, 153Weidenschilling, S. J. 1977, MNRAS, 180, 57Weidenschilling, S. J. 1977, Ap&SS, 51, 153Weidenschilling, S. J. 1980, Icar, 44, 172Weiss, L. M., Marcy, G. W., Petigura, E. A., et al. 2018, AJ, 155, 48Wisdom, J., & Holman, M. 1991, AJ, 102, 1528Xu, W., Lai, D., & Morbidelli, A. 2018, MNRAS, 481, 1538Yang, C.-C., Johansen, A., & Carrera, D. 2017, A&A, 606, A80Youdin, A. N., & Goodman, J. 2005, ApJ, 620, 459Youdin, A. N., & Lithwick, Y. 2007, Icar, 192, 588Youdin, A. N., & Shu, F. H. 2002, ApJ, 580, 494Zhang, S., Zhu, Z., Huang, J., et al. 2018, ApJL, 869, L47

23

The Astrophysical Journal, 894:143 (23pp), 2020 May 10 Batygin & Morbidelli